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Dynamics of Complex Systems
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Dynamics of Complex Systems
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Studies in Nonlinearity Series Editor: Robert L. Devaney Ralph Abraham, Dynamics: The Geometry of Behavior Ralph H. Abraham and Christopher D. Shaw, Dynamics: The Geometry of Behavior Robert L. Devaney, Chaos, Fractals, and Dynamics: Computer Experiments in Mathematics Robert L. Devaney, A First Course in Chaotic Dynamical Systems: Theory and Experiment Robert L. Devaney, An Introduction to Chaotic Dynamical Systems, Second Edition Robert L. Devaney, James F. Georges, Delbert L. Johnson, Chaotic Dynamical Systems Software Gerald A. Edgar (ed.), Classics on Fractals James Georges, Del Johnson, and Robert L. Devaney, Dynamical Systems Software Michael McGuire, An Eye for Fractals Steven H. Strogatz, Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Nicholas B. Tufillaro, Tyler Abbott, and Jeremiah Reilly, An Experimental Approach to Nonlinear Dynamics and Chaos
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Yaneer Bar-Yam
Dynamics of Complex Systems
The Advanced Book Program ▲ ▼▼
Addison-Wesley
Reading, Massachusetts
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Figure 2.4.1 ©1992 Benjamin Cummings, from E. N. Marieb/Human Anatomy and Physiology. Used with permission. Figure 7.1.1 (bottom) by Brad Smith, Elwood Linney, and the Center for In Vivo Microscopy at Duke University (A National Center for Research Resources, NIH). Used with permission. Many of the designations used by manufacturers and sellers to distinguish their products are claimed as trademarks. Where those designations appear in this book and Addison-Wesley was aware of a trademark claim, the designations have been printed in initial capital letters. Library of Congress Cataloging-in-Publication Data Bar-Yam, Yaneer. Dynamics of complex systems / Yaneer Bar-Yam. p. cm. Includes index. ISBN 0-201-55748-7 1. Biomathematics. 2. System theory. I. Title. QH323.5.B358 1997 570'.15' 1—DC21 96-52033 CIP
Copyright © 1997 by Yaneer Bar-Yam All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior written permission of the publisher. Printed in the United States of America. Addison-Wesley is an imprint of Addison Wesley Longman, Inc. Cover design by Suzanne Heiser and Yaneer Bar-Yam Text design by Jean Hammond Set in 10/12.5 Minion by Carlisle Communications, LTD 1 2 3 4 5 6 7 8 9—MA—0100999897 First printing, August 1997 Find us on the World Wide Web at http://www.aw.com/gb/
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This book is dedicated with love to my family
Zvi, Miriam, Aureet and Sageet Naomi and our children Shlomiya, Yavni, Maayan and Taeer Aureet’s memory is a blessing.
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Contents
Preface xi Acknowledgments
xv
0 Overview: The Dynamics of Complex Systems—Examples, Questions, Methods and Concepts 1
0.1
The Field of Complex Systems
1
0.2
Examples
2
0.3
Questions
6
0.4
Methods
0.5
Concepts: Emergence and Complexity
0.6
For the Instructor
8 9
14
1 Introduction and Preliminaries 16 1.1
Iterative Maps (and Chaos)
19
1.2
Stochastic Iterative Maps
1.3
Thermodynamics and Statistical Mechanics
1.4
Activated Processes (and Glasses)
1.5
Cellular Automata
1.6
Statistical Fields 145
1.7
Computer Simulations (Monte Carlo, Simulated Annealing)
1.8
Information
1.9
Computation
38 58
95
112 186
214 235
1.10 Fractals, Scaling and Renormalization
258
2 Neural Networks I: Subdivision and Hierarchy 2.1
Neural Networks: Brain and Mind
2.2
Attractor Networks
295
296
300 vii
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Contents
2.3
Feedforward Networks
322
2.4
Subdivided Neural Networks
2.5
Analysis and Simulations of Subdivided Networks
2.6
From Subdivision to Hierarchy 364
2.7
Subdivision as a General Phenomenon
328 345
366
3 Neural Networks II: Models of Mind 371 3.1
Sleep and Subdivision Training
372
3.2
Brain Function and Models of Mind
393
4 Protein Folding I: Size Scaling of Time 420 4.1
The Protein-Folding Problem
421
4.2
Introduction to the Models
4.3
Parallel Processing in a Two-Spin Model
4.4
Homogeneous Systems
4.5
Inhomogeneous Systems
4.6
Conclusions
427 432
435 458
471
5 Protein Folding II: Kinetic Pathways 472 5.1
Phase Space Channels as Kinetic Pathways
5.2
Polymer Dynamics: Scaling Theory
5.3
Polymer Dynamics: Simulations
5.4
Polymer Collapse
473
477
488
503
6 Life I: Evolution—Origin of Complex Organisms 528 6.1
Living Organisms and Environments
529
6.2
Evolution Theory and Phenomenology
6.3
Genome, Phenome and Fitness
6.4
Exploration, Optimization and Population Interactions 550
6.5
Reproduction and Selection by Resources and Predators
576
6.6
Collective Evolution: Genes, Organisms and Populations
604
6.7
Conclusions
531
542
619
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Contents
7 Life II: Developmental Biology—Complex by Design 621 7.1
Developmental Biology: Programming a Brick
621
7.2
Differentiation: Patterns in Animal Colors
7.3
Developmental Tool Kit
7.4
Theory, Mathematical Modeling and Biology
7.5
Principles of Self-Organization as Organization by Design
7.6
Pattern Formation and Evolution
626
678 688
695
8 Human Civilization I: Defining Complexity 699 8.1
Motivation
699
8.2
Complexity of Mathematical Models
8.3
Complexity of Physical Systems
8.4
Complexity Estimation
705
716
759
9 Human Civilization II: A Complex(ity) Transition 782 9.1
Introduction: Complex Systems and Social Policy
9.2
Inside a Complex System 788
9.3
Is Human Civilization a Complex System?
9.4
Toward a Networked Global Economy
9.5
Consequences of a Transition in Complexity
9.6
Civilization Itself
783
791
796 815
822
Additional Readings 827 Index 839
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Preface “Complex” is a word of the times, as in the often-quoted “growing complexity of life.” Science has begun to try to understand complexity in nature, a counterpoint to the traditional scientific objective of understanding the fundamental simplicity of laws of nature. It is believed ,h owever, that even in the study of complexity there exist simple and therefore comprehensible laws. The field of study of complex systems holds that the dynamics of complex systems are founded on universal principles that may be used to describe disparate problems ranging from particle physics to the economics of societies. A corollary is that transferring ideas and results from investigators in hitherto disparate areas will cross-fertilize and lead to important new results. In this text we introduce several of the problems of science that embody the concept of complex dynamical systems. Each is an active area of research that is at the forefront of science.Our presentation does not try to provide a comprehensive review of the research literature available in each area. Instead we use each problem as an opportunity for discussing fundamental issues that are shared among all areas and therefore can be said to unify the study of complex systems. We do not expect it to be possible to provide a succinct definition of a complex system. Instead, we give examples of such systems and provide the elements of a definition. It is helpful to begin by describing some of the attributes that characterize complex systems. Complex systems contain a large number of mutually interacting parts. Even a few interacting objects can behave in complex ways. However, the complex systems that we are interested in have more than just a few parts. And yet there is generally a limit to the number of parts that we are interested in. If there are too many parts, even if these parts are strongly interacting, the properties of the system become the domain of conventional thermodynamics—a uniform material. Thus far we have defined complex systems as being within the mesoscopic domain—containing more than a few, and less than too many parts.However, the mesoscopic regime describes any physical system on a particular length scale,and this is too broad a definition for our purposes. Another characteristic of most complex dynamical systems is that they are in some sense purposive. This means that the dynamics of the system has a definable objective or function. There often is some sense in which the systems are engineered. We address this topic directly when we discuss and contrast self-organization and organization by design. A central goal of this text is to develop models and modeling techniques that are useful when applied to all complex systems. For this we will adopt both analytic tools and computer simulation. Among the analytic techniques are statistical mechanics and stochastic dynamics.Among the computer simulation techniques are cellular automata and Monte Carlo. Since analytic treatments do not yield complete theories of complex systems, computer simulations play a key role in our understanding of how these systems work. The human brain is an important example of a complex system formed out of its component neurons. Computers might similarly be understood as complex interacting systems of transistors.Our brains are well suited for understanding complex sysxi
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tems, but not for simulating them.Why are computers better suited to simulations of complex systems? One could point to the need for precision that is the traditional domain of the computer. However, a better reason would be the difficulty the brain has in keeping track of many and arbitrary interacting objects or events—we can typically remember seven independent pieces of information at once. The reasons for this are an important part of the design of the brain that make it powerful for other purposes. The architecture of the brain will be discussed beginning in Chapter 2. The study of the dynamics of complex systems creates a host o f new interdisciplinary fields. It not only breaks down barriers between physics, chemistry and biology, but also between these disciplines and the so-called soft sciences of psychology, sociology, economics,and anthropology. As this breakdown occurs it becomes necessary to introduce or adopt a new vocabulary. Included in this new vocabulary are words that have been considered taboo in one area while being extensively used in another. These must be adopted and adapted to make them part of the interdisciplinary discourse. One example is the word “mind.” While the field of biology studies the brain,the field of psychology considers the mind.However, as the study of neural networks progresses,it is anticipated that the function of the neural network will become identified with the concept of mind. Another area in which science has traditionally been mute is in the concept of meaning or purpose. The field of science traditionally has no concept of values or valuation. Its objective is to describe natural phenomena without assigning positive or negative connotation to the description. However, the description of complex systems requires a notion of purpose,since the systems are generally purposive. Within the context of purpose there may be a concept of value and valuation. If, as we will attempt to do, we address society or civilization as a complex system and identify its purpose,then value and valuation may also become a concept that attains scientific significance. There are even further possibilities of identifying value,since the very concept of complexity allows us to identify value with complexity through its difficulty of replacement. As is usual with any scientific advance,there are both dangers and opportunities with such developments. Finally, it is curious that the origin and fate of the universe has become an accepted subject of scientific discourse—cosmology and the big bang theory—while the fate of humankind is generally the subject of religion and science fiction. There are exceptions to this rule, particularly surrounding the field of ecology—limits to population growth, global warming—however, this is only a limited selection of topics that could be addressed. Overcoming this limitation may be only a matter of having the appropriate tools. Developing the tools to address questions about the dynamics of human civilization is appropriate within the study of complex systems. It should also be recognized that as science expands to address these issues, science itself will change as it redefines and changes other fields. Different fields are often distinguished more by the type of questions they ask than the systems they study. A significant effort has been made in this text to articulate questions, though not always to provide complete answers, since questions that define the field of complex systems will inspire more progress than answers at this early stage in the development of the field.
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Like other fields, the field of complex systems has many aspects, and any text must make choices about which material to include. We have suggested that complex systems have more than a few parts and less than too many of them. There are two approaches to this intermediate regime. The first is to consider systems with more than a few parts, but still a denumerable number—denumerable,that is, by a single person in a reasonable amount of time. The second is to consider many parts, but just fewer than too many. In the first approach the main task is to describe the behavior of a particular system and its mechanism of operation—the function of a neural network of a few to a few hundred neurons, a few-celled organism, a small protein,a few people, etc. This is done by describing completely the role of each of the parts. In the second approach, the precise number of parts is not essential,and the main task is a statistical study of a collection of systems that differ from each other but share the same structure—an ensemble of systems. This approach treats general properties of proteins, neural networks, societies, etc. In this text, we adopt the second approach. However, an interesting twist to our discussion is that we will show that any complex system requires a description as a particular few-part system.A complementary volume to the present one would consider examples of systems with only a few parts and analyze their function with a view toward extracting general principles. These principles would complement the seemingly more general analysis of the statistical approach. The order of presentation of the topics in this text is a matter of taste. Many of the chapters are self-contained discussions of a particular system or question. The first chapter contains material that provides a foundation for the rest. Part of the role of this chapter is the introduction of “simple” models upon which the remainder of the text is based. Another role is the review of concepts and techniques that will be used in later chapters so that the text is more self-contained. Because of the interdisciplinary nature of the subject matter, the first chapter is considered to have particular importance. Some of the material should be familiar to most graduate students, while other material is found only in the professional literature. For example, basic probability theory is reviewed, as well as the concepts and properties of cellular automata. The purpose is to enable this text to be read by students and researchers with a variety of backgrounds. However, it should be apparent that digesting the variety of concepts after only a brief presentation is a difficult task. Additional sources of material are listed at the end of this text. Throughout the book, we have sought to limit advanced formal discussions to a minimum. When possible, we select models that can be described with a simpler formalism than must be used to treat the most general case possible. Where additional layers of formalism are particularly appropriate, reference is made to other literature. Simulations are described at a level of detail that,in most cases,should enable the student to perform and expand upon the simulations described. The graphical display of such simulations should be used as an integral part of exposure to the dynamics of these systems. Such displays are generally effective in d eveloping an intuition about what are the important or relevant properties of these systems.
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Acknowledgments This book is a composite of many ideas and reflects the efforts of many individuals that would be impossible to acknowledge. My personal efforts to compose this body of knowledge into a coherent framework for future study are also indebted to many who contributed to my own development. It is the earliest teachers, who we can no longer identify by memory, who should be acknowledged at the completion of a major effort. They and the teachers I remember from elementary school through graduate school, especially my thesis advisor John Joannopoulos, have my deepest g ratitude. Consistent with their dedication, may this be a reward for their efforts. The study of complex systems is a new endeavor, and I am grateful to a few colleagues and teachers who have inspired me to pursue this path. Charles Bennett through a few joint car trips opened my mind to the possibilities of this field and the paths less trodden that lead to it. Tom Malone, through his course on networked corporations, not only contributed significant concepts to the last chapter of this book, but also motivated the creation of my course on the dynamics of complex systems. There are colleagues and students who have inspired or contributed to my understanding of various aspects of material covered in this text. Some of this contribution arises from reading and commenting on various aspects of this text, or through discussions of the material that eventually made its way here. In some cases the discussions were originally on unrelated matters, but because they were eventually connected to these subjects,they are here acknowledged. Roughly divided by area in correspondence with the order they appear in the text these include: Glasses—David Adler; Cellular Automata—Gerard Vichniac, Tom Toffoli, Norman Margolus, Mike Biafore, Eytan Domany, Danny Kandel; Computation—Jeff Siskind; Multigrid—Achi Brandt, Shlomi Taasan, Sorin Costiner; Neural Networks—John Hopfield, Sageet Bar-Yam, Tom Kincaid, Paul Appelbaum, Charles Yang, Reza Sadr-Lahijany, Jason Redi, Lee-Peng Lee, Hua Yang, Jerome Kagan, Ernest Hartmann; Protein Folding— Elisha Haas, Charles DeLisi, Temple Smith, Robert Davenport, David Mukamel, Mehran Kardar; Polymer Dynamics—Yitzhak Rabin, Mark Smith, Boris Ostrovsky, Gavin Crooks, Eliana DeBernardez-Clark; Evolution—Alan Perelson, Derren Pierre, Daniel Goldman, Stuart Kauffman, Les Kaufman; Developmental Biology—Irving Epstein, Lee Segel, Ainat Rogel, Evelyn Fox Keller; Complexity—Charles Bennett, Michael Werman, Michel Baranger; Human Economies and Societies—Tom Malone, Harry Bloom, Benjamin Samuels, Kosta Tsipis, Jonathan King. A special acknowledgment is necessary to the students of my course from Boston University and MIT. Among them are students whose projects became incorporated in parts of this text and are mentioned above. The interest that my colleagues have shown by attending and participating in the course has brightened it for me and their contributions are meaningful: Lewis Lipsitz, Michel Baranger, Paul Barbone, George Wyner,Alice Davidson,Ed Siegel, Michael Werman,Larry Rudolfand Mehran Kardar. Among the readers of this text I am particularly indebted to the detailed comments of Bruce Boghosian, and the supportive comments of the series editor Bob Devaney. I am also indebted to the support of Charles Cantor and Jerome Kagan. xv
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I would like to acknowledge the constructive efforts of the editors at AddisonWesley starting from the initial contact with Jack Repcheck and continuing with Jeff Robbins. I thank Lynne Reed for coordinating production, and at Carlisle Communications: Susan Steines, Bev Kraus, Faye Schilling, and Kathy Davis. The software used for the text, graphs, figures and simulations of this book, includes: Microsoft Excel and Word, Deneba Canvas, Wolfram’s Mathematica, and Symantec C.The hardware includes: Macintosh Quadra,and IBM RISC workstations. The contributions of my family, to whom this book is dedicated, cannot be described in a few words. ” Yaneer Bar-Yam Newton, Massachusetts, June 1997
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0 Overview: The Dynamics of Complex Systems — Examples, Questions, Methods and Concepts
0.1
The Field of Complex Systems
The study of complex systems in a unified framework has become recognized in recent years as a new scientific discipline, the ultimate of interdisciplinary fields. It is strongly rooted in the advances that have been made in diverse fields ranging from physics to anthropology, from which it draws inspiration and to which it is relevant. Many of the systems that surround us are complex. The goal of understanding their properties motivates much if not all of scientific inquiry. Despite the great complexity and variety of systems, universal laws and phenomena are essential to our inquiry and to our understanding. The idea that all matter is formed out of the same building blocks is one of the original concepts of science. The modern manifestation of this concept—atoms and their constituent particles—is essential to our recognition of the commonality among systems in science. The universality of constituents complements the universality of mechanical laws (classical or quantum) that govern their motion. In biology, the common molecular and cellular mechanisms of a large variety of organisms form the basis of our studies.However, even more universal than the constituents are the dynamic processes of variation and selection that in some manner cause organisms to evolve. Thus, all scientific endeavor is based, to a greater or lesser degree, on the existence of universality, which manifests itself in diverse ways. In this context,the study of complex systems as a new endeavor strives to increase our ability to understand the universality that arises when systems are highly complex. A dictionary definition of the word “complex” is: “consisting of interconnected or interwoven parts.” Why is the nature of a complex system inherently related to its parts? Simple systems are also formed out of parts. To explain the difference between simple and complex systems, the terms “interconnected” or “interwoven” are somehow essential.Qualitatively, to understand the behavior of a complex system we must understand not only the behavior of the parts but how they act together to form the behavior of the whole. It is because we cannot describe the whole without describing each part, and because ea ch part must be described in relation to other parts, that complex systems are difficult to understand. This is relevant to another definition of “complex”: “not easy to understand or analyze.” These qualitative ideas about what a complex system is can be made more quantitative. Articulating them in a clear way is 1
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both essential and fruitful in pointing the way toward progress in understanding the universal properties of these systems. For many years, professional specialization has led science to progressive isolation of individual disciplines.How is it possible that well-separated fields such as molecular biology and economics can suddenly become unified in a single discipline? How does the study of complex systems in general pertain to the detailed efforts devoted to the study of particular complex systems? In this regard one must be careful to acknowledge that there is always a dichotomy between universality and specificity. A study of universal principles does not replace detailed description of particular complex systems. However, universal principles and tools guide and simplify our inquiries into the study of specifics. For the study of complex systems,universal simplifications are particularly important. Sometimes universal principles are intuitively appreciated without being explicitly stated. However, a careful articulation of such principles can enable us to approach particular systems with a systematic guidance that is often absent in the study of complex systems. A pictorial way of illustrating the relationship of the field of complex systems to the many other fields of science is indicated in Fig. 0.1.1. This figure shows the conventional view of science as progressively separating into disparate disciplines in order to gain knowledge about the ever larger complexity of systems. It also illustrates the view of the field of complex systems, which suggests that all complex systems have universal properties. Because each field develops tools for addressing the complexity of the systems in their domain, many of these tools can be adapted for more general use by recognizing their universal applicability. Hence the motivation for crossdisciplinary fertilization in the study of complex systems. In Sections 0.2–0.4 we initiate our study of complex systems by discussing examples, questions and methods that are relevant to the study of complex systems.Our purpose is to introduce the field without a strong bias as to conclusions, so that the student can develop independent perspectives that may be useful in this new field— opening the way to his or her own contributions to the study of complex systems. In Section 0.5 we introduce two key concepts—emergence and complexity—that will arise through our study of complex systems in this text.
0.2
Examples
0.2.1 A few examples What are complex systems and what properties characterize them? It is helpful to start by making a list of some examples of complex systems. Take a few minutes to make your own list. Consider actual systems rather than mathematical models (we will consider mathematical models later). Make a list of some simple things to contrast them with.
Examples of Complex Systems Governments Families The human body—physiological perspective
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(a)
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Biology
Chemistry
Mathematics
Physics
Computer Science Simple systems Philosophy Sociology Anthropology Psychology
Economics
Socio Biology logy Mat Com hemati pute cs r Sc ienc e
Psychology
s Economic
Physics
y soph Philo y olog hrop Ant
Chemistry
Complex systems
(b)
Simple systems Figure 0.1.1 Conceptual illustration of the space of scientific inquiry. (a) is the conventional view where disciplines diverge as knowledge increases because of the increasing complexity of the various systems being studied. In this view all knowledge is specific and knowledge is gained by providing more and more details. (b) illustrates the view of the field of complex systems where complex systems have universal properties. By considering the common properties of complex systems, one can approach the specifics of particular complex systems from the top of the sphere as well as from the bottom.
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A person—psychosocial perspective The brain The ecosystem of the world Subworld ecosystems: desert, rain forest, ocean Weather A corporation A computer
Examples of Simple Systems An oscillator A pendulum A spinning wheel An orbiting planet The purpose of thinking about examples is to develop a first understanding of the question, What makes systems complex? To begin to address this question we can start describing systems we know intuitively as complex and see what properties they share. We try this with the first two examples listed above as complex systems.
Government • It has many different functions:military, immigration,taxation,income distribution, transportation, regulation. Each function is itself complex. • There are different levels and types of government: local, state and federal; town meeting, council,mayoral. There are also various governmental forms in different countries.
Family • • • • •
It is a set of individuals. Each individual has a relationship with the other individuals. There is an interplay between the relationship and the qualities of the individual. The family has to interact with the outside world. There are different kinds of families: nuclear family, extended family, etc.
These descriptions focus on function and structure and diverse manifestation. We can also consider the role that time plays in complex systems. Among the properties of complex systems are change, growth and death, possibly some form of life cycle. Combining time and the environment, we would point to the ability of complex systems to adapt. One of the issues that we will need to address is whether there are different categories of complex systems. For example, we might contrast the systems we just described with complex physical systems: hydrodynamics (fluid flow, weather), glasses, composite materials, earthquakes. In what way are these systems similar to or different from the biological or social complex systems? Can we assign function and discuss structure in the same way?
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5
0.2.2 Central properties of complex systems After beginning to describe complex systems,a second step is to identify commonalities. We might make a list of some of the characteristics of complex systems and assign each of them some measure or attribute that can provide a first method of classification or description. • Elements (and their number) • Interactions (and their strength) • Formation/Operation (and their time scales) • Diversity/Variability • Environment (and its demands) • Activity(ies) (and its[their] objective[s]) This is a first step toward quantifying the properties of complex systems.Quantifying the last three in the list requires some method of counting possibilities. The problem of counting possibilities is central to the discussion of quantitative complexity.
0.2.3 Emergence: From elements and parts to complex systems There are two approaches to organizing the properties of complex systems that wil l serve as the foundation of our discussions. The first of these is the relationship between elements,parts and the whole. Since there is only one property of the complex system that we know for sure — that it is complex—the primary question we can ask about this relationship is how the complexity of the whole is related to the complexity of the parts. As we will see, this question is a compelling question for our understanding of complex systems. From the examples we have indicated above, it is apparent that parts of a complex system are often complex systems themselves. This is reasonable, because when the parts of a system are complex, it seems intuitive that a collection of them would also be complex. However, this is not the only possibility. Can we describe a system composed of simple parts where the collective behavior is complex? This is an important possibility, called emergent complexity. Any complex system formed out of atoms is an example. The idea of emergent complexity is that the behaviors of many simple parts interact in such a way that the behavior of the whole is complex.Elements are those parts of a complex system that may be considered simple when describing the behavior of the whole. Can we describe a system composed of complex parts where the collective behavior is simple? This is also possible, and it is called emergent simplicity. A useful example is a planet orbiting around a star. The behavior of the planet is quite simple, even if the planet is the Earth, with many complex systems upon it. This example illustrates the possibility that the collective system has a behavior at a different scale than its parts. On the smaller scale the system may behave in a complex way, but on the larger scale all the complex details may not be relevant.
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0.2.4 What is complexity? The second approach to the study of complex systems begins from an understanding of the relationship of systems to their descriptions. The central issue is defining quantitatively what we mean by complexity. What, after all, do we mean when we say that a system is complex? Better yet, what do we mean when we say that one system is more complex than another? Is there a way to identify the complexity of one system and to compare it with the complexity of another system? To develop a quantitative understanding of complexity we will use tools of both statistical physics and computer science—information theory and computation theory. According to this understanding, complexity is the amount of information necessary to describe a system. However, in order to arrive at a consistent definition,care must be taken to specify the level of detail provided in the description. One of our targets is to understand how this concept of complexity is related to emergence—emergent complexity and emergent simplicity. Can we understand why information-based complexity is related to the description of elements,and how their behavior gives rise to the collective complexity of the whole system? Section 0.5 of this overview discusses further the concepts of emergence and complexity, providing a simplified preview of the more complete discussions later in this text.
0.3
Questions
This text is structured around four questions related to the characterization of complex systems: 1. Space: What are the characteristics of the structure of complex systems? Many complex systems have substructure that extends all the way to the size of the system itself. Why is there substructure? 2. Time: How long do dynamical processes take in complex systems? Many complex systems have specific responses to changes in their environment that require changing their internal structure. How can a complex structure respond in a reasonable amount of time? 3. Self-organization and/versus organization by design: How do complex systems come into existence? What are the dynamical processes that can give rise to complex systems? Many complex systems undergo guided developmental processes as part of their formation. How are developmental processes guided? 4. Complexity: What is complexity? Complex systems have varying degrees of complexity. How do we characterize/distinguish the varying degrees of complexity? Chapter 1 of this text plays a special role. Its ten sections introduce mathematical tools. These tools and their related concepts are integral to our understanding of complex system behavior. The main part of this book consists of eight chapters,2–9. These
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chapters are paired.Each pair discusses one of the above four questions in the context of a particular complex system. Chapters 2 and 3 discuss the role of substructure in the context of neural networks. Chapters 4 and 5 discuss the time scale of dynamics in the context of protein folding. Chapters 6 and 7 discuss the mechanisms of organization of complex systems in the context of living organisms. Chapters 8 and 9 discuss complexity in the context of human civilization. In each case the first of the pair of chapters discusses more general issues and models. The second tends to be more specialized to the system that is under discussion. There is also a pattern to the degree of analytic, simulation or qualitative treatments. In general,the first of the two chapters is more analytic, while the second relies more on simulations or qualitative treatments. Each chapter has at least some discussion of qualitative concepts in addition to the formal quantitative discussion. Another way to regard the text is to distinguish between the two approaches summarized above. The first deals with elements and interactions. The second deals with descriptions and information. Ultimately, our objective is to relate them, but we do so using questions that progress gradually from the elements and interactions to the descriptions and information. The former dominates in earlier chapters, while the latter is important for Chapter 6 and becomes dominant in Chapters 8 and 9. While the discussion in each chapter is presented in the context of a specific complex system, our focus is on complex systems in general. Thus, we do not attempt (nor would it be possible) to review the entire fields of neural networks, protein folding, evolution, developmental biology and social and economic sciences. Since we are interested in universal aspects of these systems, the topics we cover need not be the issues of contemporary importance in the study of these systems. Our approach is to motivate a question of interest in the context of complex systems using a particular complex system, then to step back and adopt a method of study that has relevance to all complex systems. Researchers interested in a particular complex system are as likely to find a discussion of interest to them in any one of the chapters, and should not focus on the chapter with the particular complex system in its title. We note that the text is interrupted by questions that are, with few exceptions, solved in the text. They are given as questions to promote independent thought about the study of complex systems. Some of them develop further the analysis of a system through analytic work or through simulations. Others are designed for conceptual development. With few exceptions they should be considered integral to the text, and even if they are not solved by the reader, the solutions should be read.
Q
uestion 0.3.1 Consider a few complex systems. Make a list of their elements, interactions between these elements, the mechanism by which the system is formed and the activities in which the system is engaged.
Solution 0.3.1 The following table indicates properties of the systems that we will be discussing most intensively in this text. ❚
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Element
Interaction
Formation
Activity
Proteins
Amino Acids
Bonds
Protein folding
Enzymatic activity
Nervous system Neural networks
Neurons
Synapses
Learning
Behavior Thought
Physiology
Cells
Chemical messengers Physical support
Developmental biology
Movement Physiological functions
Life
Organisms
Reproduction Competition Predation Communication
Evolution
Survival Reproduction Consumption Excretion
Human economies and societies
Human Beings Communication Technology Confrontation Cooperation
Social evolution
Same as Life? Exploration?
Table 0.3.1: Complex Systems and Some Attributes
0.4
Methods
When we think about methodology, we must keep purpose in mind.Our purpose in studying complex systems is to extract general principles.General principles can take many forms. Most principles are articulated as relationships between properties— when a system has the property x, then it has the property y. When possible, relationships should be quantitative and expressed as equations. In order to explore such relationships, we must construct and study mathematical models. Asking why the property x is related to the property y requires an understanding of alternatives. What else is possible? As a bonus, when we are able to generate systems with various properties, we may also be able to use them for practical applications. All approaches that are used for the study of simple systems can be applied to the study of complex systems. However, it is important to recognize features of conventional approaches that may hamper progress in the study of complex systems. Both experimental and theoretical methods have been developed to overcome these difficulties. In this text we introduce and use methods of analysis and simulation that are particularly suited to the study of complex systems. These methods avoid standard simplifying assumptions, but use other simplifications that are better suited to our objectives. We discuss some of these in the following paragraphs. • Don’t take it apart. Since interactions between parts of a complex system are essential to understanding its behavior, looking at parts by themselves is not sufficient. It is necessary to look at parts in the context of the whole. Similarly, a complex system interacts with its environment, and this e nvironmental influence is
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important in describing the behavior of the system. Experimental tools have been developed for studying systems in situ or in vivo—in context. Theoretical analytic methods such as the mean field approach enable parts of a system to be studied in context. Computer simulations that treat a system in its entirety also avoid such problems. • Don’t assume smoothness. Much of the quantitative study of simple systems makes use of differential equations. Differential equations,like the wave equation,assume that a system is essentially uniform and that local details don’t matter for the behavior of a system on larger scales. These assumptions are not generally valid for complex systems.Alternate static models such as fractals, and dynamical models including iterative maps and cellular automata may be used instead. • Don’t assume that only a few parameters are important. The behavior of complex systems depends on many independent pieces of information. Developing an understanding of them requires us to build mental models. However, we can only have “in mind” 7±2 independent things at once. Analytic approaches, such as scaling and renormalization,have been developed to identify the few relevant parameters when this is possible. Information-based approaches consider the collection of all parameters as the object of study. Computer simulations keep track of many parameters and may be used in the study of dynamical processes. There are also tools needed for communication of the results of studies. Conventional manuscripts and oral presentations are now being augmented by video and interactive media. Such novel approaches can increase the effectiveness of communication,particularly of the results of computer simulations. However, we should avoid the “cute picture” syndrome, where pictures are presented without accompanying discussion or analysis. In this text, we introduce and use a variety of analytic and computer simulation methods to address the questions listed in the previous section. As mentioned in the preface, there are two general methods f or studying complex systems. In the first, a specific system is selected and each of the parts as well as their interactions are identified and described. Subsequently, the objective is to show how the behavior of the whole emerges from them. The second approach considers a class of systems (ensemble), where the essential characteristics of the class are described,and statistical analysis is used to obtain properties and behaviors of the systems. In this text we focus on the latter approach.
0.5
Concepts: Emergence and Complexity
The objectives of the field of complex systems are built on fundamental concepts— emergence, complexity—about which there are common misconceptions that are addressed in this section and throughout the book.Once understood,these concepts reveal the context in which universal properties of complex systems arise and specific universal phenomena, such as the evolution of biological systems, can be better understood.
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A complex system is a system formed out of many components whose behavior is emergent,that is,the behavior of the system cannot be simply inferred from the behavior of its components. The amount of information necessary to describe the behavior of such a system is a measure of its complexity. In the following sections we discuss these concepts in greater detail.
0.5.1 Emergence It is impossible to understand complex systems without recognizing that simple atoms must somehow, in large numbers, give rise to complex collective behaviors. How and when this occurs is the simplest and yet the most profound problem that the study of complex systems fa ces. The p roblem can be approached first by developing an understanding of the term “emergence.” For many, the concept of emergent behavior means that the behavior is not captured by the behavior of the parts. This is a serious misunderstanding. It arises because the collective behavior is not readily understood from the behavior of the parts. The collective behavior is, however, contained in the behavior of the parts if they are studied in the context in which they are found. To explain this,we discuss examples of emergent properties that illustrate the difference between local emergence—where collective behavior appears in a small part of the system—and global emergence—where collective behavior pertains to the system as a whole. It is the latter which is particularly relevant to the study of complex systems. We can speak about emergence when we consider a collection of elements and the properties of the collective behavior of these elements. In conventional physics, the main arena for the study of such properties is thermodynamics and statistical mechanics. The easiest thermodynamic system to think about is a gas of particles. Two emergent properties of a gas are its pressure and temperature. The reason they are emergent is that they do not naturally arise out of the description of an individual particle. We generally describe a particle by specifying its position and velocity. Pressure and temperature become relevant only when we have many particles together. While these are emergent properties, the way they are emergent is very limited. We call them local emergent properties. The pressure and temperature is a local property of the gas. We can take a very small sample of the gas away from the rest and still define and measure the (same) pressure and temperature. Such properties,called intensive in physics, are local emergent properties. Other examples from physics of locally emergent behavior are collective modes of excitation such as sound waves, or light propagation in a medium. Phase transitions (e.g., solid to liquid) also represent a collective dynamics that is visible on a macroscopic scale, but can be seen in a microscopic sample as well. Another example of a local emergent property is the formation of water from atoms of hydrogen and oxygen. The properties of water are not apparent in the properties of gasses of oxygen or hydrogen. Neither does an isolated water molecule reveal most properties of water. However, a microscopic amount of water is sufficient. In the study of complex systems we are particularly interested in global emergent properties. Such properties depend on the entire system. The mathematical treatment of global emergent properties requires some effort. This is one reason that emergence is not well appreciated or understood. We will discuss global emergence by summariz-
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ing the results of a classic mathematical treatment, and then discuss it in a more general manner that can be readily appreciated and is useful for semiquantitative analyses. The classic analysis of global emergent behavior is that of an associative memory in a simple model of neural networks known as the Hopfield or attractor network. The analogy to a neural network is useful in order to be concrete and relate this model to known concepts. However, this is more generally a model of any system formed from simple elements whose states are correlated. Without such correlations, emergent behavior is impossible. Yet if all elements are correlated in a simple way, then local emergent behavior is the outcome. Thus a model must be sufficiently rich in order to capture the phenomenon of global emergent behavior. One of the important qualities of the attractor network is that it displays global emergence in a particularly elegant manner. The following few paragraphs summarize the operation of the attractor network as an associative memory. The Hopfield network has simple binary elements that are either ON or OFF. The binary elements are an abstraction of the firing or quiescent state of neurons. The elements interact with each other to create correlations in the firing patterns. The interactions represent the role of synapses in a neural network. The network can work as a memory. Given a set o f preselected patterns, it is possible to set the interactions so that these patterns are self-consistent states of the network—the network is stable when it is in these firing patterns. Even if we change some of the neurons, the original pattern will be recovered. This is an associative memory. Assume for the moment that the pattern of firing represents a sentence, such as “To be or not to be,that is the question.”We can recover the complete sentence by presenting only part of it to the network “To be or not to be, that” might be enough. We could use any part to retrieve the whole,such as,“to be,that is the question.” This kind of memory is to be contrasted with a computer memory, which works by assigning an address to each storage location. To access the information stored in a par ticular location we need to know the address. In the neural network memory, we specify part of what is located there, rather than the analogous address: Hamlet, by William Shakespeare, act 3, scene 1, line 64. More central to our discussion,however, is that in a computer memory a particular bit of information is stored in a particular switch. By contrast,the network does not have its memory in a neuron. Instead the memory is in the synapses. In the model, there are synapses between each neuron and every other neuron. If we remove a small part of the network and look at its properties,then the number of synapses that a neuron is left with in this small part is only a small fraction of the number of synapses it started with. If there are more than a few patterns stored, then when we cut out the small part of the network it loses the ability to remember any of the patterns, even the part which would be represented by the neurons contained in this part. This kind of behavior characterizes emergent properties. We see that emergent properties cannot be studied by physically taking a system apart and looking at the parts (reductionism). They can,however, be studied by looking at each of the parts in the context of the system as a whole. This is the nature of emergence and an indication of how it can be studied and understood.
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The above discussion reflects the analysis of a relatively simple mathematical model of emergent behavior. We can,however, provide a more qualitative discussion that serves as a guide for thinking about diverse complex systems. This discussion focuses on the properties of a system when part of it is removed. Our discussion of local emergent properties suggested that taking a small part out of a large system would cause little change in the properties of the small part, or the properties of the large part.On the other hand, when a system has a global emergent property, the behavior of the small part is different in isolation than when it is part of the larger system. If we think about the system as a whole, rather than the small part of the system, we can identify the system that has a global emergent property as being formed out of interdependent parts. The term “interdependent” is used here instead of the terms “interconnected” or “interwoven” used in the dictionary definition of “complex” quoted in Section 0.1, because neither of the latter terms pertain directly to the influence one part has on another, which is essential to the properties of a dynamic system. “Interdependent” is also distinct from “interacting,” because even strong interactions do not necessarily imply interdependence of behavior. This is clear from the macroscopic properties of simple solids. Thus, we can characterize complex systems through the effect of removal of part of the system. There are two natural possibilities. The first is that properties of the part are affected, but the rest is not affected. The second is that properties of the rest are affected by the removal of a part. It is the latter that is most appealing as a model of a truly complex system. Such a system has a collective behavior that is dependent on the behavior of all of its parts. This concept becomes more precise when we connect it to a quantitative measure of complexity.
0.5.2 Complexity The second concept that is central to complex systems is a quantitative measure of how complex a system is. Loosely speaking, the complexity of a system is the amount of information needed in order to describe it. The complexity depends on the level of detail required in the description. A more formal definition can be understood in a simple way. If we have a system that could have many possible states, but we would like to specify which state it is actually in, then the number of binary digits (bits) we need to specify this particular state is related to the number of states that are possible. If we call the number of states Ω then the number of bits of information needed is log2(Ω)
I
(0.5.1)
To understand this we must realize that to specify which state the system is in, we must enumerate the states. Representing each state uniquely requires as many numbers as there are states. Thus the number of states of the representation must be the same as the number of states of the system. For a string of N bits there are 2N possible states and thus we must have Ω
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which implies that N is the same as I above. Even if we use a descriptive English text instead of numbers,there must be the same number of possible descriptions as there are states, and the information content must be the same. When the number of possible valid English sentences is properly accounted for, it turns out that the best est imate of the amount of information in English is about 1 bit per character. This means that the information content of this sentence is about 120 bits, and that of this book is about 3 106 bits. For a microstate of a physical system, where we specify the positions and momenta of each of the particles, this can be recognized as proportional to the entropy of the system, which is defined as S
k ln(Ω)
k ln(2)I
(0.5.3)
where k 1.38 10 23 Joule/˚Kelvin is the Boltzmann constant which is relevant to our conventional choice of units. Using measured entropies we find that entropies of order 10 bits per atom are typical. The reason k is so small is that the quantities of matter we typically consider are in units of Avogandro’s number (moles) and the number of bits per mole is 6.02 1023 times as large. Thus, the information in a piece of material is of order 1024 bits. There is one point about Eq.(0.5.3) that may require some clarification. The positions and momenta of particles are real numbers whose specification might require infinitely many bits. Why isn’t the information necessary to specify the microstate of a system infinite? The answer to this question comes from quantum physics, which is responsible for giving a unique value to the entropy and thus the information needed to specify a state of the system. It does this in two ways. First, it tells us that microscopic states are indistinguishable unless they differ by a discrete amount in position and momentum—a quantum diff erence given by Planck’s constant h. Second, it indicates that particles like nuclei or atoms in their ground state are uniquely specified by this state,and are indistinguishable from each other. There is no additional information necessary to specify their internal structure. Under standard conditions, essentially all nuclei are in their lowest energy state. The relationship of entropy and information is not accidental, of course, but it is the source of much confusion. The confusion arises because the entropy of a physical system is largest when it is in equilibrium. This suggests that the most complex system is a system in equilibrium. This is counter to our usual understanding of complex systems. Equilibrium systems have no spatial structure and do not change over time. Complex systems have substantial internal structure and this structure changes over time. The problem is that we have used the definition of the information necessary to specify the microscopic state (microstate) of the system rather than the macroscopic state (macrostate) of the system. We need to consider the information necessary to describe the macrostate of the system in order to define what we mean by complexity. One of the important points to realize is that in order for the macrostate of the system to require a lot of information to describe it,there must be correlations in the microstate of the system. It is only when many microscopic atoms move in a coherent fashion that we can see this motion on a macroscopic scale. However, if many
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microscopic atoms move together, the system must be far from equilibrium and the microscopic information (entropy) must be lower than that of an equilibrium system. It is helpful, even essential, to define a complexity profile which is a func tion of the scale of observation. To obtain the complexity profile, we observe the system at a particular length (or time) scale,ignoring all finer-scale details. Then we consider how much information is necessary to describe the observations on this scale. This solves the problem of distinguishing between a microscopic and a macroscopic description. Moreover, for different choices of scale, it explicitly cap tures the dependence of the complexity on the level of detail that is required in the description. The complexity profile must be a monotonically falling function of the scale.This is because the information needed to describe a system on a larger scale must be a subset of the information needed to describe the system on a smaller scale—any finerscale description contains the coarser-scale description. The complexity profile characterizes the properties of a complex system. If we wish to point to a particular number for the complexity of a system,it is natural to consider the complexity as the value of the complexity profile at a scale that is slightly smaller than the size of the system itself. The behavior at this scale includes the movement of the system through space, and dynamical changes of the system that are essentially the size of the system as a whole. The Earth orbiting the sun is a useful example. We can make a direct connection between this definition of complexity and the discussion of the formation of a complex system out of parts. The complexity of the parts of the system are described by the complexity profile of the system evaluated on the scale of the parts. When the behavior of the system depends on the behavior of the parts, the complexity of the whole must involve a description of the parts, thus it is large. The smaller the parts that must be described to describe the behavior of the whole, the larger the complexity of the entire system.
0.6
For the Instructor
This text is designed for use in an introductory graduate-level course, to present various concepts and methodologies of the study of complex systems and to begin to develop a common language for researchers in this new field. It has been used for a onesemester course, but the amount of material is large, and it is better to spread the material over two semesters.A two-semester course also provides more opportunities for including various other approaches to the study of complex systems, which are as valuable as the ones that are covered here and may be more familiar to the instructor. Consistent with the objective and purpose of the field,students attending such a course tend to have a wide variety of backgrounds and interests. While this is a positive development, it causes difficulties for the syllabus and framework of the course. One approach to a course syllabus is to include the introductory material given in Chapter 1 as an integral part of the course. It is better to interleave the later chapters with the relevant materials from Chapter 1. Such a course might proceed:1.1–1.6; 2; 3; 4; 1.7; 5; 6; 7; 1.8–1.10; 8; 9. Including the materials of Chapter 1 allows the dis-
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cussion of important mathematical methods,and addresses the diverse backgrounds of the students. Even if the introductory chapter is covered quickly (e.g., in a onesemester course),this establishes a common base of knowledge for the remainder of the course. If a high-speed approach is taken,it must be emphasized to the students that this material serves only to expose them to concepts that they are unfamiliar with, and to review concepts for those with prior knowledge of the topics covered. Unfortunately, many students are not willing to sit through such an extensive (and intense) introduction. A second approach begins from Chapter 2 and introduces the material from Chapter 1 only as needed. The chapters that are the most technically difficult,and rely the most on Chapter 1,are Chapters 4 and 5. Thus, for a one-semester course,the subject of protein folding (Chapters 4 and 5) could be skipped. Then much of the introductory material can be omitted, with the exception of a discussion of the last part of Section 1.3,and some introduction to the subject of entropy and information either through thermodynamics (Section 1.3) or information theory (Section 1.8), preferably both. Then Chapters 2 and 3 can be covered first, followed by Chapters 6–9, with selected material introduced from Chapter 1 as is appropriate for the background of the students. There are two additional recommendations.First,it is better to run this course as a project-based course rather than using graded homework. The varied backgrounds of students make it difficult to select and fairly grade the problems. Projects for individuals or small groups of students can be tailored to their knowledge and interests. There are many new areas of inquiry, so that projects may approach research-level contributions and be exciting for the students. Unfortunately, this means that students may not devote sufficient effort to the study of course material,and rely largely upon exposure in lectures. There is no optimal solution to this problem. Second,if it is possible,a seminar series with lecturers who work in the field should be an integral part of the course. This provides additional exposure to the varied approaches to the study of complex systems that it is not possible for a single lecturer or text to provide.
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1 Introduction and Preliminaries Conceptual Outline ❚ 1.1 ❚
A deceptively simple model of the dynamics of a system is a deterministic iterative map applied to a single real variable. We characterize the dynamics by looking at its limiting behavior and the approach to this limiting behavior. Fixed points that attract or repel the dynamics, and cycles, are conventional limiting behaviors of a simple dynamic system. However, changing a parameter in a quadratic iterative map causes it to undergo a sequence of cycle doublings (bifurcations) until it reaches a regime of chaotic behavior which cannot be characterized in this way. This deterministic chaos reveals the potential importance of the influence of fine-scale details on large-scale behavior in the dynamics of systems.
❚ 1.2 ❚
A system that is subject to complex (external) influences has a dynamics that may be modeled statistically. The statistical treatment simplifies the complex unpredictable stochastic dynamics of a single system, to the simple predictable dynamics of an ensemble of systems subject to all possible influences. A random walk on a line is the prototype stochastic process. Over time, the random influence causes the ensemble of walkers to spread in space and form a Gaussian distribution. When there is a bias in the random walk, the walkers have a constant velocity superimposed on the spreading of the distribution.
❚ 1.3 ❚
While the microscopic dynamics of physical systems is rapid and complex, the macroscopic behavior of many materials is simple, even static. Before we can understand how complex systems have complex behaviors, we must understand why materials can be simple. The origin of simplicity is an averaging over the fast microscopic dynamics on the time scale of macroscopic observations (the ergodic theorem) and an averaging over microscopic spatial variations. The averaging can be performed theoretically using an ensemble representation of the physical system that assumes all microscopic states are realized. Using this as an assumption, a statistical treatment of microscopic states describes the macroscopic equilibrium behavior of systems. The final part of Section 1.3 introduces concepts that play a central role in the rest of the book. It discusses the differences between equilibrium and complex systems. Equilibrium systems are divisible and satisfy the ergodic theorem. Complex systems
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are composed out of interdependent parts and violate the ergodic theorem. They have many degrees of freedom whose time dependence is very slow on a microscopic scale.
❚ 1.4 ❚
To understand the separation of time scales between fast and slow degrees of freedom, a two-well system is a useful model. The description of a particle traveling in two wells can be simplified to the dynamics of a two-state (binary variable) system. The fast dynamics of the motion within a well is averaged by assuming that the system visits all states, represented as an ensemble. After taking the average, the dynamics of hopping between the wells is represented explicitly by the dynamics of a binary variable. The hopping rate depends exponentially on the ratio of the energy barrier and the temperature. When the temperature is low enough, the hopping is frozen. Even though the two wells are not in equilibrium with each other, equilibrium continues to hold within a well. The cooling of a two-state system serves as a simple model of a glass transition, where many microscopic degrees of freedom become frozen at the glass transition temperature.
❚ 1.5 ❚
Cellular automata are a general approach to modeling the dynamics of spatially distributed systems. Expanding the notion of an iterative map of a single variable, the variables that are updated are distributed on a lattice in space. The influence between variables is assumed to rely upon local interactions, and is homogeneous. Space and time are both discretized, and the variables are often simplified to include only a few possible states at each site. Various cellular automata can be designed to model key properties of physical and biological systems.
❚ 1.6 ❚
The equilibrium state of spatially distributed systems can be modeled by fields that are treated using statistical ensembles. The simplest is the Ising model, which captures the simple cooperative behavior found in magnets and many other systems. Cooperative behavior is a mechanism by which microscopic fast degrees of freedom can become slow collective degrees of freedom that violate the ergodic theorem and are visible macroscopically. Macroscopic phase transitions are the dynamics of the cooperative degrees of freedom. Cooperative behavior of many interacting elements is an important aspect of the behavior of complex systems. This should be contrasted to the two-state model (Section 1.4), where the slow dynamics occurs microscopically.
❚ 1.7 ❚
Computer simulations of models such as molecular dynamics or cellular automata provide important tools for the study of complex systems. Monte Carlo simulations enable the study of ensemble averages without necessarily describing the dynamics of a system. However, they can also be used to study random-walk dynamics. Minimization methods that use iterative progress to find a local minimum are often an important aspect of computer simulations. Simulated annealing is a method that can help find low energy states on complex energy surfaces.
❚ 1.8 ❚
We have treated systems using models without acknowledging explicitly that our objective is to describe them. All our efforts are designed to map a system onto a description of the system. For complex systems the description must be quite long, and the study of descriptions becomes essential. With this recognition, we turn
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to information theory. The information contained in a communication, typically a string of characters, may be defined quantitatively as the logarithm of the number of possible messages. When different messages have distinct probabilities P in an ensemble, then the information can be identified as ln(P ) and the average information is defined accordingly. Long messages can be modeled using the same concepts as a random walk, and we can use such models to estimate the information contained in human languages such as English.
❚ 1.9 ❚
In order to understand the relationship of information to systems, we must also understand what we can infer from information that is provided. The theory of logic is concerned with inference. It is directly linked to computation theory, which is concerned with the possible (deterministic) operations that can be performed on a string of characters. All operations on character strings can be constructed out of elementary logical (Boolean) operations on binary variables. Using Turing’s model of computation, it is further shown that all computations can be performed by a universal Turing machine, as long as its input character string is suitably constructed. Computation theory is also related to our concern with the dynamics of physical systems because it explores the set of possible outcomes of discrete deterministic dynamic systems.
❚ 1.10 ❚
We return to issues of structure on microscopic and macroscopic scales by studying fractals that are self-similar geometric objects that embody the concept of progressively increasing structure on finer and finer length scales. A general approach to the scale dependence of system properties is described by scaling theory. The renormalization group methodology enables the study of scaling properties by relating a model of a system on one scale with a model of the system on another scale. Its use is illustrated by application to the Ising model (Section 1.6), and to the bifurcation route to chaos (Section 1.1). Renormalization helps us understand the basic concept of modeling systems, and formalizes the distinction between relevant and irrelevant microscopic parameters. Relevant parameters are the microscopic parameters that can affect the macroscopic behavior. The concept of universality is the notion that a whole class of microscopic models will give rise to the same macroscopic behavior, because many parameters are irrelevant. A conceptually related computational technique, the multigrid method, is based upon representing a problem on multiple scales.
The study of complex systems begins from a set of models that capture aspects of the dynamics of simple or complex systems. These models should be sufficiently general to encompass a wide range of possibilities but have sufficient structure to capture interesting features. An exciting bonus is that even the apparently simple mo dels discussed in this chapter introduce features that are not typically treated in the conventional science of simple systems, but are appropriate introductions to the dynamics of complex systems.Our treatment of dynamics will often consider discrete rather than continuous time. Analytic treatments are often convenient to formulate in continu-
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ous variables and differential equations;however, computer simulations are often best formulated in discrete space-time variables with well-defined intervals. Moreover, the assumption of a smooth continuum at small scales is not usually a convenient starting point for the study of complex systems. We are also generally interested not only in one example of a system but rather in a class of systems that differ from each other but share a characteristic structure. The elements of such a class of systems are collectively known as an ensemble. As we introduce and study mathematical models, we should recognize that our primary objective is to represent properties of real systems. We must therefore develop an understanding of the nature of models and modeling, and how they can pertain to either simple or complex systems.
1.1
Iterative Maps (and Chaos)
An iterative map f is a function that evolves the state of a system s in discrete time s(t) = f(s(t − t))
(1.1.1)
where s(t) describes the state of the system at time t. For convenience we will generally measure time in units of t which then has the value 1,and time takes integral values starting from the initial condition at t = 0. Many of the complex systems we will consider in this text are of the form of Eq .( 1 . 1 . 1 ) ,i f we allow s to be a general variable of arbitrary dimension. The generality of iterative maps is discussed at the end of this section. We start by considering several examples of iterative maps where s is a single variable. We discuss briefly the binary variable case, s = ±1. Then we discuss in greater detail two types of maps with s a real variable, s ∈ℜ, linear maps and quadratic maps. The quadratic iterative map is a simple model that can display complex dynamics. We assume that an iterative map may be started at any initial condition allowed by a specified domain of its system variable.
1.1.1 Binary iterative maps There are only a few binary iterative maps.Question 1.1.1 is a complete enumeration of them.*
Q
uestion 1.1.1 Enumerate all possible iterative maps where the system is described by a single binary variable, s = ±1.
Solution 1.1.1 There are only four possibilities: s(t) = 1 s(t) = −1
(1.1.2)
s(t) = s(t − 1) s(t) = −s(t − 1)
*Questions are an integral part of the text. They are designed to promote independent thought. The reader is encouraged to read the question, contemplate or work out an answer and then read the solution provided in the text. The continuation of the text assumes that solutions to questions have been read.
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It is instructive to consider these possibilities in some detail. The main reason there are so few possibilities is that the form of the iterative map we are using depends,at most, on the value of the system in the previous time. The first two examples are constants and don’t even depend on the value of the system at the previous time. The third map can only be distinguished from the first two by observation of its behavior when presented with two different initial conditions. The last of the four maps is the only map that has any sustained dynamics. It cycles between two values in perpetuity. We can think about this as representing an oscillator. ❚
Q
uestion 1.1.2
a. In what way can the map s(t) = −s(t − 1) represent a physical oscillator? b. How can we think of the static map, s(t) = s(t − 1), as an oscillator? c. Can we do the same for the constant maps s(t) = 1 and s(t) = −1? Solution 1.1.2 (a) By looking at the oscillator displacement with a strobe at half-cycle intervals,our measured values can be represented by this map. (b) By looking at an oscillator with a strobe at cycle intervals. (c) You might think we could, by picking a definite starting phase of the strobe with respect to the oscillator. However, the constant map ignores the first value, the oscillator does not. ❚
1.1.2 Linear iterative maps: free motion, oscillation, decay and growth The simplest example of an iterative map with s real, s ∈ℜ, is a constant map: s(t) = s0
(1.1.3)
No matter what the initial value,this system always takes the particular value s0. The constant map may seem trivial,however it will be useful to compare the constant map with the next class of maps. A linear iterative map with unit coefficient is a model of free motion or propagation in space: s(t) = s(t − 1) + v
(1.1.4)
at successive times the values of s are separated by v, which plays the role of the velocity.
Q
uestion 1.1.3 Consider the case of zero velocity s(t) = s(t − 1)
(1.1.5)
How is this different from the constant map? Solution 1.1.3 The two maps differ in their dependence on the initial value. ❚
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Runaway growth or decay is a multiplicative iterative map: s(t) = gs(t − 1)
(1.1.6)
We can generate the values of this iterative map at all times by using the equivalent expression s(t ) = g t s 0 = e ln(g )t s 0
(1.1.7)
which is exponential growth or decay. The iterative map can be thought of as a sequence of snapshots of Eq.(1.1.7) at integral time. g = 1 reduces this map to the previous case.
Q
uestion 1.1.4 We have seen the case of free motion, and now jumped to the case of growth. What happened to accelerated motion? Usually we would consider accelerated motion as the next step after motion with a constant velocity. How can we write accelerated motion as an iterative map? Solution 1.1.4 The description of accelerated motion requires two variables: position and velocity. The iterative map would look like: x(t) = x(t − 1) + v(t − 1) v(t) = v(t − 1) + a
(1.1.8)
This is a two-variable iterative map. To write this in the notation of Eq.(1.1.1) we would define s as a vector s(t) = (x(t), v(t)). ❚
Q
uestion 1.1.5 What happens in the rightmost exponential expression in Eq. (1.1.7) when g is negative?
Solution 1.1.5 The logarithm of a negative number results in a phase i . The term i t in the exponent alternates sign every time step as one would expect from Eq. (1.1.6). ❚ At this point,it is convenient to introduce two graphical methods for describing an iterative map. The first is the usual way of plotting the value of s as a function of time. This is shown in the left panels of Fig. 1.1.1. The second type of plot,shown in the right panels, has a different purpose. This is a plot of the iterative relation s(t) as a function of s(t − 1). On the same axis we also draw the line for the identity map s(t) = s(t − 1). These two plots enable us to graphically obtain the successive values of s as follows. Pick a starting value of s, which we can call s(0). Mark this value on the abscissa. Mark the point on the graph of s(t) that corresponds to the point whose abscissa is s(0),i.e.,the point (s(0), s(1)).Draw a horizontal line to intersect the identity map. The intersection point is (s(1), s(1)). Draw a vertical line back to the iterative map. This is the point (s(1), s(2)). Successive values of s(t) are obtained by iterating this graphical procedure. A few examples are plotted in the right panels of Fig. 1.1.1. In order to discuss the iterative maps it is helpful to recognize several features of these maps.First,intersection points of the identity map and the iterative map are the fixed points of the iterative map: s 0 = f (s 0 )
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s(t)
(a) s(t)
s(t)=c s(t)=c
0
1
(b) s(t)
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Figure 1.1.1 The left panels show the time-dependent value of the system variable s(t) resulting from iterative maps. The first panel (a) shows the result of iterating the constant map; (b) shows the result of adding v to the previous value during each time interval; (c)–(f) show the result of multiplying by a constant g, where each figure shows the behavior for a different range of g values: (c) g > 1, (d) 0 < g < 1, (e) 1 < g < 0, and (f) g < 1. The right panels are a different way of showing graphically the results of iterations and are constructed as follows. First plot the function f(s) (solid line), where s(t) f(s(t 1)). This can be thought of as plotting s(t) vs. s(t 1). Second, plot the identity map s(t) s(t 1) (dashed line). Mark the initial value s(0) on the horizontal axis, and the point on the graph of s(t) that corresponds to the point whose abscissa is s(0), i.e. the point (s(0), s(1)). These are shown as squares. From the point (s(0), s(1)) draw a horizontal line to intersect the identity map. The intersection point is (s(1), s(1)). Draw a vertical line back to the iterative map. This is the point (s(1), s(2)). Successive values of s(t) are obtained by iterating this graphical procedure. ❚
Fixed points,not surprisingly, play an important role in iterative maps. They help us describe the state and behavior of the system after many iterations. There are two kinds of fixed points—stable and unstable. Stable fixed points are characterized by “attracting” the result of iteration of points that are nearby. More precisely, there exists
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a neighborhood of points of s0 such that for any s in this neighborhood the sequence of points (1.1.10) {s, f (s), f 2(s), f 3 (s),…} converges to s0. We are using the notation f 2(s) = f(f (s)) for the second iteration,and similar notation for higher iterations. This sequence is just the time series of the iterative map for the initial condition s. Unstable fixed points have the opposite behavior, in that iteration causes the system to leave the neighborhood of s0. The two types of fixed points are also called attracting and repelling fixed points. The family of multiplicative iterative maps in Eq.(1.1.6) all have a fixed point at s0 = 0. Graphically from the figures, or analytically from Eq. (1.1.7), we see that the fixed point is stable for |g| < 1 and is unstable for |g| > 1. There is also distinct behavior of the system depending on whether g is positive or negative. For g < 0 the iterations alternate from one side to the other of the fixed point, whether it is attracted to or repelled from the fixed point. Specifically, if s < s0 then f(s) > s0 and vice versa, or sign(s − s0) = −sign(f (s) − s0). For g > 0 the iteration does not alternate.
Q
uestion 1.1.6 Consider the iterative map. s(t) = gs(t − 1) + v
(1.1.11)
convince yourself that v does not affect the nature of the fixed point, only shifts its position.
Q
uestion 1.1.7 Consider an arbitrary iterative map of the form Eq.(1.1.1), with a fixed point s0 (Eq.(1.1.9)). If the iterative map can be expanded in a Taylor series around s0 show that the first derivative g=
df (s) ds s
(1.1.12) 0
characterizes the fixed point as follows: For |g | < 1, s0 is an attracting fixed point. For |g| > 1, s0 is a repelling fixed point. For g < 0, iterations alternate sides in a sufficiently small neighborhood of s0. For g > 0 ,i terations remain on one side in a sufficiently small neighborhood of s0. Extra credit: Prove the same theorem for a differentiable function (no Taylor expansion needed) using the mean value theorem. Solution 1.1.7 If the iterative map can be expanded in a Taylor series we write that f (s) = f (s 0 ) + g (s − s 0 )+ h (s − s 0 )2 +…
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where g is the first derivative at s0, and h is one-half of the second derivative at s0. Since s0 is a fixed point f(s0) = s0 we can rewrite this as: f (s) −s 0 = g +h (s − s 0 ) +… s − s0
(1.1.14)
If we did not have any higher-order terms beyond g, then by inspection each of the four conditions that we have to prove would follow from this expression without restrictions on s. For example, if |g| > 1, then taking the magnitude of both sides shows that f(s) − s0 is larger than s − s0 and the iterations take the point s away from s0. If g > 0,then this expression says that f(s) stays on the same side of s0. The other conditions follow similarly. To generalize this argument to include the higher-order terms of the expansion, we must guarantee that whichever domain g is in (g > 1, 0 < g < 1, −1 < g < 0, or g < −1), the same is also true of the whole right side. For a Taylor expansion, by choosing a small enough neighborhood |s − s0| < , we can guarantee the higher-order terms are less than any number we choose. We choose to be half of the minimum of |g − 1|, |g − 0| and |g + 1|. Then g + is in the same domain as g. This provides the desired guarantee and the proof is complete. We have proven that in the vicinity of a fixed point the iterative map may be completely characterized by its first-order expansion (with the exception of the special points g = ±1,0). ❚ Thus far we have not considered the special cases g =±1,0. The special cases g = 0 and g = 1 have already been treated as simpler iterative maps. When g = 0, the fixed point at s = 0 is so attractive that it is the result of any iteration. When g = 1 all points are fixed points. The new special case g = −1 has a different significance. In this case all points alternate between positive and negative values, repeating every other iteration. Such repetition is a generalization of the fixed point. Whereas in the fixed-point case we repeat every iteration, here we repeat after every two iterations. This is called a 2-cycle, and we can immediately consider the more general case of an n-cycle. In this terminology a fixed point is a 1-cycle.One way to describe an n-cycle is to say that iterating n times gives back the same result, or equivalently, that a new iterative map which is the nth fold composition of the original map h = f n has a fixed point. This description would include also fixed points of f and all points that are m-cycles, where m is a divisor of n. These are excluded from the definition of the n-cycles. While we have introduced cycles using a map where all points are 2-cycles,more general iterative maps have specific sets of points that are n-cycles. The set of points of an n-cycle is called an orbit. There are a variety of properties of fixed points and cycles that can be proven for an arbitrary map. One of these is discussed in Question 1.1.8.
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Q
uestion 1.1.8 Prove that there is a fixed point between any two points of a 2-cycle if the iterating function f is continuous.
Solution 1.1.8 Let the 2-cycle be written as s 2 = f (s1 )
(1.1.15)
s1 = f (s 2 )
Consider the function h(s) = f(s) − s, h(s1) and h(s2) have opposite signs and therefore there must be an s0 between s1 and s2 such that h(s0) = 0—the fixed point. ❚ We can also generalize the definition of attracting and repelling fixed points to consider attracting and repelling n-cycles. Attraction and repulsion for the cycle is equivalent to the attraction and repulsion of the fixed point of f n.
1.1.3 Quadratic iterative maps: cycles and chaos The next iterative map we will consider describes the effect of nonlinearity (self-action): s(t) = as(t − 1)(1 − s(t − 1))
(1.1.16)
f (s) = as(1 − s)
(1.1.17)
or equivalently This map has played a significant role in development of the theory of dynamical systems because even though it looks quite innocent,it has a dynamical behavior that is not described in the conventional science of simple systems. Instead, Eq. (1.1.16) is the basis of significant work on chaotic behavior, and the transition of behavior from simple to chaotic. We have chosen this form of quadratic map because it simplifies somewhat the discussion. Question 1.1.11 describes the relationship between this family of quadratic maps,parameterized by a, and what might otherwise appear to be a different family of quadratic maps. We will focus on a values in the range 4 > a > 0. For this range, any value of s in the interval s ∈[0,1] stays within this interval. The minimum value f(s) = 0 occurs for s = 0,1 and the maximal value occurs for s = 1/2. For all values of a there is a fixed point at s = 0 and there can be at most two fixed points, since a quadratic can only intersect a line (Eq. (1.1.9)) in two points. Taking the first derivative of the iterative map gives df = a(1− 2s) ds
(1.1.18)
At s = 0 the derivative is a which, by Question 1.1.7,shows that s = 0 is a stable fixed point for a < 1 and an unstable fixed point for a > 1. The switching of the stability of the fixed point at s = 0 coincides with the introduction of a second fixed point in the interval [0,1] (when the slope at s = 0 is greater than one, f (s) > s for small s, and since
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f (1) = 0, we have that f(s1) = s1 for some s1 in [0,1] by the same construction as in Question 1.1.8). We find s1 by solving the equation s1 = as1 (1− s1 )
(1.1.19)
s1 = (a −1)/a
(1.1.20)
Substituting this into Eq. (1.1.18) gives df = 2− a ds s 1
(1.1.21)
This shows that for 1 < a < 3,the new fixed point is stable by Question 1.1.7. Moreover, the derivative is positive for 1 < a < 2,so s1 is stable and convergence is from one side. The derivative is negative for 2 < a < 3, so s1 is stable and alternating. Fig. 1.1.2(a)–(c) shows the three cases: a = 0.5, a = 1.5 and a = 2.8. For a = 0.5, starting from anywhere within [0,1] leads to convergence to s = 0. When s(0) > 0.5 the first iteration takes the system to s(1) < 0.5. The closer we start to s(0) = 1 the closer to s = 0 we get in the first jump. At s(0) = 1 the convergence to 0 occurs in the first jump. A similar behavior would be found for any value of 0 < a < 1. For a = 1.5 the behavior is more complicated. Except for the points s = 0,1,the convergence is always to the fixed point s1 = (a − 1)/a between 0 and 1. For a = 2.8 the iterations converge to the same point;however, the convergence is alternating. Because there can be at most two fixed points for the quadratic map, one might think that this behavior would be all that would happen for 1 < a < 4.One would be wrong. The first indication that this is not the case is the instability of the fixed point at s1 starting from a = 3. What happens for a > 3? Both of the fixed points that we have found,and the only ones that can exist for the quadratic map, are now unstable. We know that the iteration of the map has to go somewhere, and only within [0,1]. The only possibility, within our experience, is that there is an attracting n-cycle to which the fixed points are unstable. Let us then consider the map f 2(s) whose fixed points are 2-cycles of the original map. f 2(s) is shown in the right panels of Fig. 1.1.2 for increasing values of a. The fixed points of f (s) are also fixed points of f 2(s). However, we see that two additional fixed points exist for a > 3. We can also show analytically that two fixed points are introduced at exactly a = 3: f 2(s) = a 2s(1 −s)(1 − as(1 − s))
(1.1.22)
To find the fixed point we solve: s = a 2s(1− s)(1 −as(1 −s))
(1.1.23)
We already know two solutions of this quartic equation—the fixed points of the map f. One of these at s = 0 is obvious. Dividing by s we have a cubic equation: a3s 3 − 2a 3s 2 + a2 (1 + a)s + (1 −a 2 ) = 0
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Figure 1.1.2 (pp. 28-30) Plots of the result of iterating the quadratic map f(s) = as(1 − s) for different values of a. The left and center panels are similar to the left and right panels of Fig. 1.1.1. The left panels plot s(t). The center panels describe the iteration of the map f (s) on axes corresponding to s(t) and s(t − 1). The right panels are similar to the center panels but are for the function f 2(s). The different values of a are indicated on the panels and show the changes from (a) convergence to s = 0 for a = 0.5, (b) convergence to s = (a − 1)/ a for a = 1.5, (c) alternating convergence to s = (a − 1)/ a for a = 2.8, (d) bifurcation — convergence to a 2-cycle for a = 3.2, (e) second bifurcation — convergence to a 4-cycle for a = 3.5, (f) chaotic behavior for a = 3.8. ❚
We can reduce the equation to a quadratic by dividing by (s − s1) as follows (we simplify the algebra by dividing by a(s − s1) = (as − (a − 1))): a2 s 2 −a(a + 1)s +(a +1)
)
(as −(a −1)) a3 s 3 −2a3s 2 +a 2 (1 + a)s +(1 −a 2 ) a3 s 3 −(a −1)a2 s 2 −(a + 1)a 2s 2 + a 2 (1 + a)s +(1 −a 2 ) −(a +1)a 2s 2 + a(1 + a)(a − 1)s
(1.1.25)
+ a(1 + a)s +(1 −a 2 ) Now we can obtain the roots to the quadratic: a 2s 2 − a(a + 1)s +(a +1) = 0
(1.1.26)
(a +1) ± (a +1)(a − 3) 2a
(1.1.27)
s2 =
This has two solutions (as it must for a 2-cycle) for a <−1 or for a > 3. The former case is not of interest to us since we have assumed 0 < a < 4. The latter case is the two roots that are promised. Notice that for exactly a = 3 the two roots that are the new 2-cycle are the same as the fixed point we have already found s1. The 2-cycle splits off from the fixed point at a = 3 when the fixed point becomes unstable. The two attracting points continue to separate as a increases. For a > 3 we expect that the result of iteration eventually settles down to the 2-cycle. The system state alternates between the two roots Eq. (1.1.27). This is shown in Fig. 1.1.2(d). As we continue to increase a beyond 3, the 2-cycle will itself become unstable at a point that can be calculated by setting df 2 ds
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= −1
(1.1.28)
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to be a = 1 + √6 = 3.44949. At this value of a the 2-cycle splits into a 4-cycle (Fig. 1.1.2(e)).Each of the fixed points of f 2(s) simultaneously split into 2-cycles that together form a 4-cycle for the original map.
Q
uestion 1.1.9 Show that when f has a 2-cycle, both of the fixed points of f 2 must split simultaneously.
Solution 1.1.9 The split occurs when the fixed points become unstable— the derivative of f 2 equals –1. We can show that the derivative is equal at the two fixed points of Eq. (1.1.27), which we call s ±2: df 2 ds
= s2
df (f (s)) df (s) df (s) = ds s ds f (s ) ds s 2 2 2
(1.1.29)
where we have made use of the chain rule.Since f (s2+) = s2− and vice versa, we have shown this expression is the same whether s2 = s+2 or s2 = s2−. Note: This can be generalized to show that the derivative of f k is the same at all of its k fixed points corresponding to a k-cycle of f. ❚ The process of taking an n-cycle into a 2n-cycle is called bifurcation.Bifurcation continues to replace the limiting behavior of the iterative map with progressively longer cycles of length 2k. The bifurcations can be simulated. They occur at smaller and smaller intervals and there is a limit point to the bifurcations at ac = 3.56994567. Fig. 1.1.3 shows the values that are reached by the iterative map at long times—the stable cycles—as a function of a < ac . We will discuss an algebraic treatment of the bifurcation regime in Section 1.10. Beyond the bifurcation regime a > ac (Fig. 1.1.2(f)) the behavior of the iterative map can no longer be described using simple cycles that attract the iterations. The behavior in this regime has been identified with chaos. Chaos has been characterized in many ways, but one property is quite generally agreed upon—the inherent la ck of predictability of the system dynamics. This is often expressed more precisely by describing the sensitivity of the system’s fate to the initial conditions.A possible definition is: There exists a distance d such that for any neighborhood V of any point s it is possible to find a point s′ within the neighborhood and a number of iterations k so that f k(s′) is further than d away from f k(s). This means that arbitrarily close to any point is a point that will be displaced a significant distance away by iteration. Qualitatively, there are two missing aspects of this definition,first that the points that move far away must not be too unlikely (otherwise the system is essentially predictable) and second that d is not too small (in which case the divergence of the dynamics may not be significant). If we look at the definition of chaotic behavior, we see that the concept of scale plays an important role.A small distance between s and s′ turns into a large distance between f k(s) and f k(s′). Thus a fine-scale difference eventually becomes a large-scale difference. This is the essence of chaos as a model of complex system behavior. To understand it more fully, we can think about the state variable s not as one real variable,
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Figure 1.1.3 A plot of values of s visited by the quadratic map f(s) = as(1 − s) after many iterations as a function of a, including stable points, cycles and chaotic behavior. The different regimes are readily apparent. For a < 1 the stable point is s = 0. For 1 < a < 3 the stable point is at s0 = (a − 1)/a. For 3 < a < ac with ac = 3.56994567, there is a bifurcation cascade with 2-cycles then 4-cycles, etc. 2k-cycles for all values of k appear in progressively narrower regions of a. Beyond 4-cycles they cannot be seen in this plot. For a > ac there is chaotic behavior. There are regions of s values that are not visited and regions that are visited in the long time behavior of the quadratic map in the chaotic regime which this figure does not fully illustrate. ❚
but as an infinite sequence of binary variables that form its binary representation s = 0.r1r2r3r4 ... Each of these binary variables represents the state of the system—the value of some quantity we can measure about the system—on a particular length scale. The higher order bits represent the larger scales and the lower order ones represent the finer scales. Chaotic behavior implies that the state of the first few binary variables, r1r2, at a particular time are determined by the value of fine scale variables at an earlier time. The farther back in time we look, the finer scale variables we have to consider in order to know the present values of r1r2. Because many different variables are relevant to the behavior of the system, we say that the system has a complex behavior. We will return to these issues in Chapter 8. The influence of fine length scales on coarse ones makes iterative maps difficult to simulate by computer. Computer representations of real numbers always have finite precision. This must be taken into account if simulations of iterative maps or chaotic complex systems are performed.
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Another significant point about the iterative map as a model of a complex system is that there is nothing outside of the system that is influencing it.All of the information we need to describe the behavior is contained in the precise value of s. The complex behavior arises from the way the different parts of the system—the fine and course scales—affect each other.
Q
uestion 1.1.10: Why isn’t the iterative map in the chaotic regime equivalent to picking a number at random?
Solution 1.1.10: We can still predict the behavior of the iterative map over a few iterations. It is only when we iterate long enough that the map becomes unpredictable. More specifically, the continuity of the function f (s) guarantees that for s and s′ close together f (s) and f (s′) will also be close together. Specifically, given an it is possible to find a such that for |s − s′| < , | f (s)− f (s′)| < . For the family of functions we have been considering, we only need to set < /a, since then we have: | f (s) − f (s ′ )| = a | s(1− s)− s ′(1 − s ′ )| = a | s − s ′ ||1 −(s + s ′ )| < a |s − s ′ | < (1.1.30) Thus if we fix the number of cycles to be k, we can always find two points close enough so that | f k(s′)−f k(s)|< by setting | s − s′|< /ak. ❚ The tuning of the parameter a leading from simple convergent behavior through cycle bifurcation to chaos has been identified as a universal description of the appearance of chaotic behavior from simple behavior of many systems. How do we take a complicated real system and map it onto a discrete time iterative map? We must define a system variable and then take snapshots of it at fixed intervals (or at least welldefined intervals). The snapshots correspond to an iterative map. Often there is a natural choice for the interval that simplifies the iterative behavior. We can then check to see if there is bifurcation and chaos in the real system when parameters that control the system behavior are varied. One of the earliest examples of the application of iterative maps is to the study of heart attacks. Heart attacks o ccur in many different ways. One kind of heart attack is known as fibrillation. Fibrillation is characterized by chaotic and ineffective heart muscle contractions. It has been suggested that bifurcation may be observed in heartbeats as a period doubling (two heartbeats that are inequivalent). If correct, this may serve as a warning that the heart structure, due to various changes in heart tissue parameters, may be approaching fibrillation. Another system where more detailed studies have suggested that bifurcation occurs as a route to chaotic behavior is that of turbulent flows in hydrodynamic systems.A subtlety in the application of the ideas of bifurcation and chaos to physical systems is that physical systems are better modeled as having an increasing number of degrees of freedom at finer scales. This is to be contrasted with a system modeled by a single real number, which has the same number of degrees of freedom (represented by the binary variables above) at each length scale.
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1.1.4 Are all dynamical systems iterative maps? How general is the iterative map as a tool for describing the dynamics of systems? There are three apparent limitations of iterative maps that we will consider modifying later, Eq. (1.1.1): a. describes the homogeneous evolution of a system since f itself does not depend on time, b. describes a system where the state of the system at time t depends only on the state of the system at time t – t, and c. describes a deterministic evolution of a system. We can,however, bypass these limitations and keep the same form of the iterative map if we are willing to let s describe not just the present state of the system but also a. the state of the system and all other factors that might affect its evolution in time, b. the state of the system at the present time and sufficiently many previous times, and c. the probability that the system is in a particular state. Taking these caveats together, all of the systems we will consider are iterative maps, which therefore appear to be quite general.Generality, however, can be quite useless, since we want to discard as much information as possible when describing a system. Another way to argue the generality of the iterative map is through the laws of classical or quantum dynamics.If we consider s to be a variable that describes the positions and velocities of all particles in a system, all closed systems described by classical mechanics can be described as deterministic iterative maps.Quantum evolution of a closed system may also be described by an iterative map if s describes the wave function of the system. However, our intent is not necessarily to describe microscopic dynamics, but rather the dynamics of variables that we consider to be relevant in describing a system. In this case we are not always guaranteed that a deterministic iterative map is sufficient. We will discuss relevant generalizations, first to stochastic maps, in Section 1.2.
E
xtra Credit Question 1.1.11 Show that the system of quadratic iterative maps s(t ) = s(t −1)2 + k
(1.1.31)
is essentially equivalent in its dynamical properties to the iterative maps we have considered in Eq. (1.1.16). Solution 1.1.11 Two iterative maps are equivalent in their properties if we can perform a time-independent one-to-one map of the time-dependent system states from one case to the other. We will attempt to transform the family of quadratic maps given in this problem to the one of Eq.(1.1.16) using a linear map valid at all times s(t ) = ms ′(t ) +b
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By direct substitution this leads to: ms ′(t )+ b = (ms ′(t − 1)+ b)2 +k
(1.1.33)
We must now choose the values of m and b so as to obtain the form of Eq. (1.1.16). s ′(t ) = ms ′(t −1)(s ′(t −1)+
2b 1 ) + (b 2 +k − b) m m
(1.1.34)
For a correct placement of minus signs in the parenthesis we need: 2b 1 s ′(t ) = (−m)s ′(t −1)( − − s ′(t − 1)) + (b 2 +k − b) m m
(1.1.35)
b2 −b +k = 0
(1.1.36)
2b = −1 m
(1.1.37)
b = (1 ± 1 − 4k )/2
(1.1.38)
a = −m = 2b = (1 ± 1− 4k )
(1.1.39)
or
giving
We see that for k < 1/4 we have two solutions. These solutions give all possible (positive and negative) values of a. What about k > 1/4? It turns out that this case is not very interesting compared to the rich behavior for k < 1/4, since there are no finite fixed points,and therefore by Question 1.1.8 no 2-cycles (it is not hard to generalize this to n-cycles). To c onfirm this, verify that iterations diverge to +∞ from any initial condition. Note: The system of equations of this question are the ones extensively analyzed by Devaney in his excellent textbook A First Course in Chaotic Dynamical Systems. ❚
E
xtra Credit Question 1.1.12 You are given a problem to solve whic h when reduced to mathematical form looks like s = f c (s)
(1.1.40)
where f is a complicated function that depends on a parameter c. You know that there is a solution of this equation in the vicinity of s0. To solve this equation you try to iterate it (Newton’s method) and it works,since you find that f k(s0) converges nicely to a solution. Now, however, you realize that you need to solve this problem for a slightly different value of the parameter c, and when you try to iterate the equation you can’t get the value of s to converge. Instead the values start to oscillate and then behave in a completely erratic
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way. Suggest a solution for this problem and see if it works for the function fc (s) = cs(1 − s), c = 3.8, s0 = 0.5. A solution is given in stages (a) - (c) below. Solution 1.1.12(a) A common resolution of this problem is to consider iterating the function: hc (s) = s +(1 − )f c (s) where we can adjust
(1.1.41)
to obtain rapid convergence. Note that solutions of s = hc (s)
(1.1.42)
are the same as solutions of the original problem.
Q
uestion 1.1.12(b) Explain why this could work.
Solution 1.1.12(b) The derivative of this function at a fixed point can be controlled by the value of . It is a linear interpolation between the fixed point derivative of fc and 1. If the fixed point is unstable and oscillating, the derivative of fc must be less than −1 and the interpolation should help. We can also explain this result without appealing to our work on iterative maps by noting that if the iteration is causing us to overshoot the mark, it makes sense to mix the value s we start from with the value we get from fc(s) to get a better estimate.
Q
uestion 1.1.12(c) Explain how to pick .
Solution 1.1.12(c) If the solution is oscillating, then it makes sense to assume that the fixed point is in between successive values and the distance is revealed by how much further it gets each time;i.e., we assume that the iteration is essentially a linear map near the fixed point and we adjust so that we compensate exactly for the overshoot of fc . Using two trial iterations, a linear approximation to fc at s0 looks like: s 2 = f c (s 1) ≈ g(s 1 − s 0 )+ s 0 s3 = fc (s 2 ) ≈ g(s 2 −s 0 ) +s 0
(1.1.43)
Adopting the linear approximation as a definition of g we have: g ≡ (s 3 − s2 )/(s 2 − s1)
(1.1.44)
Set up so that the first it eration of the modified system will take you to the desired answer: s 0 = s 1 + (1− )fc (s1 )
(1.1.45)
s 0 −s1 =(1 − )(f c (s 1) − s1 ) = (1− )(s 2 − s1 )
(1.1.46)
(1 − ) =(s 0 − s1 )/(s 2 −s 1 )
(1.1.47)
or
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To eliminate the unknown s0 we use Eq. (1.1.43) to obtain: (s2 − s1 ) = g(s1 − s 0 )+ (s0 − s1 )
(1.1.48)
(s0 − s1 ) = (s 2 −s 1 )/(1 − g )
(1.1.49)
or = 1/(1− g)
1−
(1.1.50)
= −g /(1 − g ) = (s 2 − s 3 )/(2s 2 − s1 −s 3 )
(1.1.51)
It is easy to check, using the formula in terms of g, that the modified iteration has a zero derivative at s0 when we use the approximate linear forms for fc . This means we have the best convergence possible using the information from two iterations of fc . We then use the value of to iterate to convergence. Try it! ❚
1.2
Stochastic Iterative Maps
Many of the systems we would like to consider are described by system variables whose value at the next time step we cannot predict with complete certainty. The uncertainty may arise from many sources,including the existence of interactions and parameters that are too complicated or not very relevant to our problem. We are then faced with describing a system in which the outcome of an iteration is probabilistic and not deterministic. Such systems are called stochastic systems. There are several ways to describe such systems mathematically. One of them is to consider the outcome of a particular update to be selected from a set of possible values. The probability of each of the possible values must be specified. This description is not really a model of a single system, because each realization of the system will do something different. Instead,this is a model of a collection of systems—an ensemble.Our task is to study the properties of this ensemble. A stochastic system is generally described by the time evolution of random variables. We begin the discussion by defining a random variable.A random variable s is defined by its probability distribution Ps(s′), which describes the likelihood that s has the value s′. If s is a continuous variable,then Ps (s′)ds′ is the probability that s resides between s′ and s′ + ds′. Note that the subscript is the variable name rather than an index. For example, s might be a binary variable that can have the value +1 or −1. Ps (1) is the probability that s = 1 and Ps (−1) is the probability that s = −1. If s is the outcome of an unbiased coin toss, with heads called 1 and tails called −1, both of these values are 1/2.When no confusion can arise,the notation Ps (s′) is abbreviated to P(s), where s may be either the variable or the value. The sum over all possible values of the probability must be 1.
∑ Ps (s ′) =1
(1.2.1)
s′
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In the discussion of a system described by random variables, we often would like to know the average value of some quantity Q(s) that depends in a definite way on the value of the stochastic variable s. This average is given by: < Q(s) >=
∑ Ps (s ′)Q(s ′) s′
(1.2.2)
Note that the average is a linear operation. We now consider the case of a time-dependent random variable. Rather than describing the time dependence of the variable s(t), we describe the time dependence of the probability distribution Ps (s′;t). Similar to the iterative map, we can consider the case where the outcome only depends on the value of the system variable at a previous time,and the transition probabilities do not depend explicitly on time. Such systems are called Markov chains. The transition probabilities from a state at a particular time to the next discrete time are written: Ps (s ′(t )| s ′(t −1))
(1.2.3)
Ps is used as the notation for the transition probability, since it is also the probability distribution of s at time t, given a particular value s′(t − 1) at the previous time. The use of a time index for the arguments illustrates the use of the transition probability. Ps (1|1) is the probability that when s = 1 at time t − 1 then s = 1 at time t. Ps (−1|1) is the probability that when s = 1 at time t − 1 then s =−1 at time t. The transition probabilities,along with the initial probability distribution of the system Ps (s′; t = 0), determine the time-dependent ensemble that we are interested in. Assuming that we don’t lose systems on the way, the transition probabilities of Eq. (1.2.3) must satisfy:
∑ Ps (s ′′ | s ′) = 1
(1.2.4)
s ′′
This states that no matter what the value of the system variable is at a particular time, it must reach some value at the next time. The stochastic system described by transition probabilities can be written as an iterative map on the probability distribution P(s) Ps (s ′;t ) =
∑ Ps (s ′ |s ′′)Ps (s ′′;t −1) s′
(1.2.5)
It may be more intuitive to write this using the notation Ps (s ′(t );t ) =
∑ Ps (s ′(t )|s ′(t − 1))Ps (s ′(t −1);t −1)
s ′(t −1)
(1.2.6)
in which case it may be sufficient, though hazardous, to write the abbreviated form P(s(t )) =
∑ P(s(t )|s(t −1))P(s(t −1))
s(t −1)
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It is important to recognize that the time evolution equation for the probability is linear. The linear evolution of this system (Eq. (1.2.5)) guarantees that superposition applies. If we start with an initial distribution at time t = 0, then we could find the result at time t by separately looking at the evolution of each of the probabilities P 1(s;0) and P 2(s ;0). Explicitly we can write P(s;t) = P 1(s ;t) + P 2(s;t). The meaning of this equation should be well noted. The right side of the equation is the sum of the evolved probabilities P 1(s ;t) and P 2(s;t). This linearity is a direct consequence of the independence of different members of the ensemble and says nothing about the complexity of the dynamics. We note that ultimately we are interested in the behavior of a particular system s(t) that only has one value of s at every time t. The ensemble describes how many such systems will behave. Analytically it is easier to describe the ensemble as a whole,however, simulations may also be used to observe the behavior of a single system.
1.2.1 Random walk Stochastic systems with only one binary variable might seem to be trivial, but we will devote quite a bit of attention to this problem. We begin by considering the simplest possible binary stochastic system. This is the system which corresponds to a coin toss. Ideally, for each toss there is equal probability of heads (s = +1) or tails (s = −1), and there is no memory from one toss to the next. The ensemble at each time is independent of time and has an equal probability of ±1: P(s;t) = 12
1 s,1 + 2
(1.2.8)
s ,−1
where the discrete delta function is defined by 1
i ,j
=
0
i =j i≠j
(1.2.9)
Since Eq. (1.2.8) is independent of what happens at all previous times, the evolution of the state variable is given by the same expression P(s ′ |s) = 12
s ′ ,1
+ 12
(1.2.10)
s ′,−1
We can illustrate the evaluation of the average of a function of s at time t: < Q(s) >t =
∑Q(s ′)Ps (s ′;t ) = ∑ Q(s ′)( 12
s ′= ±1
s ′ =±1
s ′ ,1
+ 12
s ′ ,−1
)=
1 2
∑Q(s ′) (1.2.11)
s ′ = ±1
For example, if we just take Q(s) to be s itself we have the average of the system variable: < s >t = 12
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(1.2.12)
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Q
uestion 1.2.1 Will you win more fair coin tosses if (a) you pick heads every time, or if (b) you alternate heads and tails, or if (c) you pick heads or tails at random or if (d) you pick heads and tails by some other system? Explain why. Solution 1.2.1 In general, we cannot predict the number of coin tosses that will be won, we can only estimate it based on the chance of winning. Assuming a fair coin means that this is the best that can be done. Any of the possibilities (a)–(c) give the same chance of winning. In none of these ways of gambling does the choice you make correlate with the result of the coin toss. The only system (d) that can help is if you have some information about what the result of the toss will be,like betting on the known result after the coin is tossed.A way to write this formally is to write the probability distribution of the choice that you are making. This choice is also a stochastic process. Calling the choice c(t), the four possibilities mentioned are: (a) P(c;t ) =
(1.2.13)
c,1 t
(b) P(c;t ) = 1+(−1) 2 (c) P(c;t ) = (d) P(c;t ) =
1 2 c,1
t
c,1
+ 12
+ 1 −(−1) 2
c,−1
= mod2 (t )
c,1
+ mod2 (t +1)
c,−1
(1.2.14) (1.2.15)
c,−1
(1.2.16)
c,s(t )
It is sufficient to show that the average probability of winning is the same in each of (a)–(c) and is just 1/2. We follow through the manipulations in order to illustrate some concepts in the treatment of more than one stochastic variable. We have to sum over the probabilities of each of the possible values of the coin toss and each of the values of the choices, adding up the probability that they coincide at a particular time t: <
c,s
>=
∑∑ s′
c′
c ′ ,s ′ Ps (s ′;t )Pc (c ′ ,t )
(1.2.17)
This expression assumes that the values of the coin toss and the value of the choice are independent,so that the joint probability of having a particular value of s and a particular value of c is the product of the probabilities of each of the variables independently: Ps,c (s ′ , c ′ ;t) = Ps (s ′;t )Pc (c ′;t ) (1.2.18) —the probabilities-of-independent-variables factor. This is valid in cases (a)–(c) and not in case (d), where the probability of c occurring is explicitly a function of the value of s. We evaluate the probability of winning in each case (a) through (c) using
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<
c,s
∑′ ∑′ c′,s ′ (12 s ′,1 + 12 s ′,−1)Pc (c ′;t ) s c = ∑ ( 21 c ′,1 + 12 c ′,−1 )Pc (c ′ ;t ) c′ = ∑ (12 Pc (1;t ) + 12 Pc (−1;t )) = 12
>=
(1.2.19)
c′
where the last equality follows from the normalization of the probability (the sum over all possibilities must be 1, Eq. (1.2.1)) and does not depend at all on the distribution. This shows that the independence of the variables guarantees that the probability of a win is just 1/2. For the last case (d) the trivial answer, that a win is guaranteed by this method of gambling, can be arrived at formally by evaluating <
c,s
>=
∑∑ s′
c′
c ′ ,s ′ Ps ,c (s ′ , c ′;t )
(1.2.20)
The value of s at time t is independent of the value of c, but the value of c depends on the value of s. The joint probability Ps,c(s′,c′;t) may be written as the product of the probability of a particular value of s = s′ times the conditional probability Pc(c′|s′;t) of a particular value of c = c′ given the assumed value of s: <
c,s
>=
∑∑
c ′ ,s ′ Ps (s ′;t )Pc (c ′ | s ′ ;t )
∑s ′ ∑ c′
c ′, s ′Ps ( s ′;t ) c ′ , s ′
s′
=
c′
=
Ps (s ′ ;t ) = 1 ∑ s′
(1.2.21) ❚
The next step in our analysis of the binary stochastic system is to consider the behavior of the sum of s(t) over a particular number of time steps. This sum is the difference between the total number of heads and the total number of tails. It is equivalent to asking how much you will win or lose if you gamble an equal amount of money on each coin toss after a certain number of bets. This problem is known as a random walk, and we will define it as a consideration of the state variable t
d(t ) =
∑ s(t ′)
(1.2.22)
t ′ =1
The way to write the evolution of the state variable is: P(d ′ |d) = 12
1 d ′,d+1 + 2
d ′,d −1
(1.2.23)
Thus a random walk considers a state variable d that can take integer values d ∈ { . . . , − 1,0,1, . . . }. At every time step, d(t) can only move to a value one higher or one lower than where it is. We assume that the probability of a step to the right (higher) is equal to that of a step to the left (lower). For convenience, we assume (with no loss of gener-
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43
ality) that the system starts at position d(0) = 0. This is built into Eq.(1.2.22). Because of the symmetry of the system under a shift of the origin, this is equivalent to considering any other starting point. Once we solve for the probability distribution of d at time t, because of superposition we can also find the result of evolving any initial probability distribution P(d;t = 0). We can picture the random walk as that of a drunk who has difficulty consistently moving forward. Our model of this walk assumes that the drunk is equally likely to take a step forward or backward. Starting at position 0, he moves to either +1 or −1. Let’s say it was +1. Next he moves to +2 or back to 0. Let’s say it was 0. Next to +1 or −1. Let’s say it was +1. Next to +2 or 0. Let’s say +2. Next to +3 or +1. Let’s say +1. And so on. What is the value of system variable d(t) at time t? This is equivalent to asking how far has the walk progressed after t steps.Of course there is no way to know how far a particular system goes without watching it. The average distance over the ensemble of systems is the average over all possible values of s(t). This average is given by applying Eq. (1.2.2) or Eq. (1.2.11) to all of the variables s(t): < d(t ) > = 1
2
=
∑
K1
s(t )=±1 t
2
∑
s (3)=±1
1 2
∑
s (2)=±1
1 2
∑
d(t )
s(1)=±1
∑ < s(t ′) > = 0
(1.2.24)
t ′ =1
The average is written out explicitly on the first line using Eq.(1.2.11). The second line expression can be arrived at either directly or from the linearity of the average. The final answer is clear, since it is equally likely for the walker to move to the right as to the left. We can also ask what is a typical distance traveled by a particular walker. By typical distance we mean how far from the starting point. This can either be defined by the average absolute value of the distance, or as is more commonly accepted,the root mean square (RMS) distance: (1.2.25)
(t ) = < d(t )2 > 2
t t t < d(t ) > = < s(t ′) > = < s(t ′)s(t ′′) > = < s(t ′ )s(t ′′ ) > t ′=1 t ′ ,t ′′ =1 t ′ ,t ′′ =1 2
∑
∑
∑
(1.2.26)
To evaluate the average of the product of the two steps, we treat differently the case in which they are the same step and when they are different steps. When the two steps are the same one we use s(t) = ±1 to obtain: < s(t)2 > = <1 > =1
(1.2.27)
Which follows from the normalization of the probability (or is obvious). To evaluate the average of the product of two steps at different times we need the joint probability of s(t ) and s(t ′). This is the probability that each of them will take a particular
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value. Because we have assumed that the steps are independent, the joint probability is the product of the probabilities for each one separately: P(s(t ),s(t ′ )) = P(s(t ))P(s(t ′))
t ≠ t′
(1.2.28)
so that for example there is 1/4 chance that s(t) = +1 and s(t) = −1. The independence of the two steps leads the average of the product of the two steps to factor: < s(t)s(t ′) > =
∑ P(s(t ),s(t ′ ))s(t)s(t ′) ∑ P(s(t ))P(s(t ′ ))s(t )s(t ′)
s (t ), s(t ′ )
=
t ≠ t′
(1.2.29)
s (t ),s(t ′ )
= < s(t ) > < s(t ′ ) > =0 This is zero, since either of the averages are zero. We have the combined result: < s(t)s(t ′) > =
(1.2.30)
t ,t ′
and finally: < d(t )2 > =
t
t
∑ < s(t ′)s(t ′′) > = ∑
t ′ ,t ′′ =1
t ′ ,t ′′ =1
t
t ′,t ′
=
∑1 = t
(1.2.31)
t ′=1
This gives the classic and important result that a random walk travels a typical distance that grows as the square root of the number of steps taken: . We can now consider more completely the probability distribution of the position of the walker at time t. The probability distribution at t = 0 may be written: P(d ;0) =
(1.2.32)
d,0
After the first time step the probability distribution changes to P(d ;1) =
1 2 d,1
+ 12
(1.2.33)
d,−1
this results from the definition d(1) = s (1). After the second step d(2) = s(1) + s(2) it is: P(d ;2) =
1 4 d,2
+ 12
d,0
+ 14
d,−2
(1.2.34)
More generally it is not difficult to see that the probabilities are given by normalized binomial coefficients,since the number of ones chosen out of t steps is equivalent to the number of powers of x in (1 + x)t. To reach a position d after t steps we must take (t + d)/2 steps to the right and (t − d)/2 steps to the left. The sum of these is the number of steps t and their difference is d. Since each choice has 1/2 probability we have:
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P(d ,t ) =
t 1 t (d + t )/2 2
t! 2 [(d + t )/2]![(t − d)/2]! (1 +(−1)t +d ) = 2
oddeven t ,d oddeven t ,d
=
1 t
45
oddeven t ,d
(1.2.35)
where the unusual delta function imposes the condition that d takes only odd or only even values depending on whether t is odd or even. Let us now consider what happens after a long time.The probability distribution spreads out,and a single step is a small distance compared to the typical distance traveled. We can consider s and t to be continuous variables where both conditions d,t >> 1 are satisfied. Moreover, we can also consider d<< t, because the chance that all steps will be taken in one direction b ecomes very small. This enables us to use Sterling’s approximation to the factorial x!~ 2 x e −x x x
(1.2.36)
ln(x!)~ x(lnx −1) + ln( 2 x )
For large t it also makes sense not to restrict d to be either odd or even. In order to allow both, we,in effect, interpolate and then take only half of the probability we have in Eq. (1.2.35). This leads to the expression: P(d ,t ) = =
t t e −t
t
2 (t −d)(t +d )2t [(d + t )/2][(d+t )/2][(t − d)/2][(t −d)/ 2]e −(d+t )/2−(t −d)/2 (2 t(1− x 2 )) −1/ 2 (1 + x)[(1+x )t / 2](1 − x)[(1−x )t / 2] (1.2.37)
where we have defined x = d / t. To approximate this expression it is easier to consider it in logarithmic form: ln(P(d,t ))= −(t /2)[(1+ x)ln(1 + x) +(1 − x)ln(1 − x)]−(1/2)ln(2 t(1 − x 2 )) ≈ −(t /2)[(1 + x)(x − x 2 /2 +K) +(1 − x)(−x − x 2 /2 + K)]− (1/2)ln(2 t + K) = −tx 2 /2 − ln( 2 t )
(1.2.38)
or exponentiating: P(d ,t ) =
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1
e −d
2
2 t
/ 2t
=
1
e −d
2
/2
2
2
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The prefactor of the exponential, 1/√2 , originates from the factor √2 x in Eq. (1.2.36). It is independent of d and takes care of the normalization of the probability. The result is a Gaussian distribution. Questions 1.2.2–1.2.5 investigate higherorder corrections to the Gaussian distribution.
Q
uestion 1.2.2 In order to obtain a correction to the Gaussian distribution we must add a correction term to Sterling’s approximation: x!~ 2 x e − x x x (1+
1 +…) 12x
ln(x!)~ x(lnx −1) + ln( 2 x ) + ln(1 +
1 +…) 12x
(1.2.40)
Using this expression, find the first correction term to Eq. (1.2.37). Solution 1.2.2 The correction term in Sterling’s approximation contributes a factor to Eq. (1.2.37) which is (for convenience we write here c = 1/12): (1+ c /t ) 3c 1 = (1− + …) =(1 − +…) (1.2.41) (1 + 2c /(t + d))(1 + 2c /(t − d)) t 4t where we have only kept the largest correction term,neglecting d compared to t. Note that the correction term vanishes as t becomes large. ❚
Q
uestion 1.2.3 Keeping additional terms of the expansion in Eq.(1.2.38), and the result of Question 1.2.2,find the first order correction terms to the Gaussian distribution.
Solution 1.2.3 Correction terms in Eq. (1.2.38) arise from several places. We want to keep all terms that are of order 1/t. To do this we must keep in mind that a typical distance traveled is d ∼ √t, so that . The next terms are obtained from: ln(P(d,t ))= −(t /2)[(1+ x)ln(1 + x) +(1 − x)ln(1 − x)] − (1/2)ln(2 t(1 − x 2 ))+ ln(1− 1/4t ) ≈ −(t /2)[(1+ x)(x − 1 x 2 + 1 x 3 − 1 x 4 K) 2
3
4
+ (1 − x)(−x − 12 x 2 − 31 x 3 − 41 x 4 K)] − ln( 2 t ) −(1/2)ln(1− x 2 ) + ln(1− 1/4t) ≈ −(t /2)[(x + x 2 − 1 x 2 − 1 x 3 + 1 x 3 + 1 x 4 − 1 x 4 K) 2
2
3
3
(1.2.42)
4
+ (−x + x 2 − 1 x 2 + 1 x 3 − 1 x 3 + 1 x 4 − 1 x 4 K)] 2
2
3
3
4
− ln( 2 t ) +(x 2 /2 +…)+ (−1/4t + …) = −tx 2 /2 − ln( 2 t ) −tx 4 /12 + x 2 /2 − 1/4t This gives us a distribution:
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P(d ,t ) =
1 2 t
e −d
2
4
3
2
2
/ 2t −d / 12t +d / 2t −1 / 4t
e
47
(1.2.43) ❚
Q
uestion 1.2.4 What is the size of the additional factor? Estimate the size of this term as t becomes large.
Solution 1.2.4 The typical value of the variable d is its root mean square value = √t. At this value the additional term gives a factor e1 / 6t
(1.2.44)
which approaches 1 as time increases. ❚
Q
uestion 1.2.5 What is the fraction error that we will make if we neglect this term after one hundred steps? After ten thousand steps?
Solution 1.2.5 After one hundred time steps the walker has traveled a typical distance of ten steps. We generally approximate the probability of arriving at this distance using Eq.(1.2.39). The fractional error in the probability of arriving at this distance according to Eq. (1.2.44) is 1 − e1/6t ≈ −1 / 6t = −0.00167. So already at a distance of ten steps the error is less than 0.2%. It is much less likely for the walker to arrive at the distance 2 = 20. The ratio of the probability to arrive at 20 compared to 10 is e−2 / e−0.5 ∼ 0.22. If we want to know the error of this smaller probability case we would write (1 − e−16/12t +4/2t−1/4t) = (1 − e5/12t ) ≈ −0.0042, which is a larger but still small error. After ten thousand steps the errors are smaller than the errors at one hundred steps by a factor of one hundred. ❚
1.2.2 Generalized random walk and the central limit theorem We can generalize the random walk by allowing a variety of steps from the current location of the walker to sites nearby, not only to the adjacent sites and not only to integer locations. If we restrict ourselves to steps that on average are balanced left and right and are not too long ranged, we can show that all such systems have the same behavior as the simplest random walk at long enough times (and characteristically not even for very long times). This is the content of the central limit theorem. It says that summing any set of independent random variables eventually leads to a Gaussian distribution of probabilities, which is the same distribution as the one we arrived at for the random walk. The reason that the same distribution arises is that successive iteration of the probability update equation, Eq.(1.2.7),smoothes out the distribution, and the only relevant information that survives is the width of the distribution which is given by (t). The proof given below makes use of a Fourier transform and can be skipped by readers who are not well acquainted with transforms. In the next section we will also include a bias in the random walk. For long times this can be described as
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Introduction and Preliminaries
an average motion superimposed on the unbiased random walk.We start with the unbiased random walk. Each step of the random walk is described by the state variable s(t) at time t. The probability of a particular step size is an unspecified function that is independent of time: P(s;t) = f (s)
(1.2.45)
We treat the case of integer values of s. The continuum case is Question 1.2.6. The absence of bias in the random walk is described by setting the average displacement in a single step to zero: <s > =
∑ sf (s) = 0
(1.2.46)
s
The statement above that each step is not too long ranged,is mathematically just that the mean square displacement in a single step has a well-defined value (i.e., is not infinite): < s2 > =
∑ s 2 f (s) =
2 0
(1.2.47)
s
Eqs. (1.2.45)–(1.2.47) hold at all times. We can still evaluate the average of d(t) and the RMS value of d(t) directly using the linearity of the average: t
< d(t ) > = <
∑ s(t ′ ) > =t < s > = 0
(1.2.48)
t ′ =1
2
t t < d(t ) > = < s(t ′) > = < s(t ′)s(t ′′) > t ′=1 t ′ ,t ′′ =1
∑
2
∑
(1.2.49)
Since s(t′) and s(t″) are independent for t′ ≠ t′′, as in Eq. (1.2.29), the average factors: < s(t ′)s(t ′′) > = <s(t ′) >< s(t ′′) > = 0
t′ ≠ t′′
(1.2.50)
2 0
(1.2.51)
Thus, all terms t′ ≠ t′′ are zero by Eq. (1.2.46). We have: < d(t )2 > =
t
∑ < s(t ′)2 > = t t ′=1
This means that the typical value of d(t) is 0√t . To obtain the full distribution of the random walk state variable d(t) we have to sum the stochastic variables s(t). Since d(t) = d(t − 1) + s(t) the probability of transition from d(t − 1) to d(t) is f (d(t) − d(t − 1)) or:
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P(d ′ |d) = f (d ′ −d)
49
(1.2.52)
We can now write the time evolution equation and iterate it t times to get P (d;t). P(d;t ) =
∑ P(d | d ′)P(d ′ ;t − 1) = ∑ f (d − d ′)P(d ′ ;t − 1)
(1.2.53)
d'
d'
This is a convolution, so the most convenient way to effect a t fold iteration is in Fourier space. The Fourier representation of the probability and transition functions for integral d is:
∑ e −ikd P(d;t )
P˜(k;t ) ≡ f˜(k) ≡
d
∑ e −iks f (s)
(1.2.54)
s
We use a Fourier series because of the restriction to integer values of d. Once we solve the problem using the Fourier representation, the probability distribution is recovered from the inverse formula: 1 2
P(d ;t ) =
∫ dke ikd P˜(k ;t )
(1.2.55)
−
which is proved 1 2
∫ dke
−
1 = 2
ikd
1 P˜(k;t) = 2
∑ P(d ′;t ) ∫ dke d′
ikd e −ikd ′ P(d ′ ;t ) ∫ dke ∑ d′
−
ik(d−d ′ )
=
∑ P(d ′;t ) d,d′ = P(d ;t)
(1.2.56)
d′
−
using the expression: d,d ′
=
1 2
∫ dke ik(d−d ′)
(1.2.57)
−
Applying Eq. (1.2.54) to Eq. (1.2.53):
∑ e −ikd ∑ f (d −d ′)P(d ′;t − 1) d d′ = ∑ ∑e −ik(d−d ′) e −ikd ′ f (d − d ′)P(d ′;t −1) d′ d = ∑ ∑e −ikd ′′e −ikd ′ f (d ′′ )P(d ′;t −1) d′ d ′′ = ∑ e −ikd ′′ f (d ′′)∑ e −ikd ′P(d ′;t −1) = f˜(k)P˜(k;t − 1)
P˜(k;t ) =
d′
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we can iterate the equation to obtain: P˜(k;t ) = f˜(k) P˜(k ;t −1) = f˜(k)t
(1.2.59)
where we use the definition d(1) = s(1) that ensures that P(d;1) = P(s;1) = f (d). For large t the walker has traveled a large distance,so we are interested in variations of the probability P(d;t) over large distances.Thus,in Fourier space we are con˜ near cerned with small values of k. To simplify Eq.(1.2.59) for large t we expand f(k) ˜ k = 0. From Eq.(1.2.54) we can directly evaluate the derivatives of f (k) at k = 0 in terms of averages: dn f˜(k) dn k
=
∑(−is)n f (s) = (−i)n < s n >
(1.2.60)
s
k =0
We can use this expression to evaluate the terms of a Taylor expansion of f˜(k): f˜(k) 1 f˜(k) = f˜(0) + k+ k k =0 2
2˜
f (k)
k
k 2 +K
2
(1.2.61)
k =0
1 f˜(k) =< 1 > −i < s >k − < s 2 > k 2 + K 2
(1.2.62)
Using the normalization of the probability (< 1 > = 1),and Eqs.(1.2.46) and (1.2.47), gives us:
(
P˜(k;t ) = 1 − 21
2 2 0k
)
t
+K
(1.2.63)
We must now remember that a typical value of d(t) ,f rom its RMS value, is 0√t. By the properties of the Fourier transform,this implies that a typical value of k that we must consider in Eq.(1.2.63) varies with time as 1/√t. The next term in the expansion,cubic in k, would give rise to a term that is smaller by this factor, and therefore becomes unimportant at long times. If we write k = q /√t, then it becomes clearer how to write Eq. (1.2.63) using a limiting expression for large t : P˜(k;t ) = 1 − 21
2 2 0q
t
t
+K ~e −
2 2 0q
/2
= e −t
2 2 0k / 2
(1.2.64)
This Gaussian, when Fourier transformed back to an expression in d, gives us a Gaussian as follows: P(d ;t ) =
1 2
∫
dke ikd e −t
2 2 0k
−
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1 2
∞
∫ dke ikde −t
2 2 0k
/2
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51
We have extended the integral because the decaying exponential becomes narrow as t increases. The integral is performed by completing the square in the exponent, giving: 1 = 2
∞
2
∫ dke −d / 2t
2 0
e −(t
2 2 2 0 k −2ikd−d
/t
2 0
)/2
1
=
2 t
−∞
2 0
e −d
2
/ 2t
2 0
(1.2.66)
or equivalently: 1
P(d ;t ) =
(t )2
2
e −d
2
/ 2 (t )
2
(1.2.67)
which is the same as Eq. (1.2.39).
Q
uestion 1.2.6 Prove the central limit theorem when s takes a continuum of values.
Solution 1.2.6 The proof follows the same course as the integer valued case. We must define the appropriate averages,and the transform. The average of s is still zero, and the mean square displacement is defined similarly:
∫
< s > = ds sf (s) = 0
∫
< s 2 > = ds s 2 f (s) =
(1.2.46´) 2 0
(1.2.47´)
To avoid problems of notation we substitute the variable x for the state variable d: t
< x(t) > = <
∑ s(t ′) > = t < s > = 0
(1.2.48´)
t ′ =1
Skipping steps that are the same we find: 2
t t < x(t) > = < s(t ′) > = < s(t ′ )2 > = t t ′ =1 t ′=1
∑
2
∑
2 0
(1.2.51´)
since s(t′) and s(t′′) are still independent for t′ ≠ t′′. Eq. (1.2.53) is also essentially unchanged:
∫
P(x;t) = dx ′ f (x − x ′ )P(x ′ ;t −1)
(1.2.53´)
The transform and inverse transform must now be defined using
∫ ∫
P˜(k;t ) ≡ dx e −ikx P(x ;t ) ˜f (k) ≡ ds e −iks f (s)
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P(d ;t ) =
1 2
∫ dke ikd P˜(k ;t )
(1.2.55´)
The latter is proved using the properties of the Dirac (continuum) delta function: (x − x ′) =
1 2
∫ dke ik(x −x ) ′
(1.2.56´)
∫ dx ′ (x − x ′)g(x ′) = g(x) where the latter equation holds for an arbitrary function g(x). The remainder of the derivation carries forward unchanged. ❚
1.2.3 Biased random walk We now return to the simple random walk with binary steps of ±1. The model we consider is a random walk that is biased in one direction.Each time a step is taken there is a probability P+ for a step of +1, that is different from the probability P– for a step of –1, or: P(s;t) = P+
s ,1
P(d ′ |d) = P+
+ P−
d ′,d +1
(1.2.68)
s ,−1
+ P−
d ′,d−
(1.2.69)
where P+ + P– = 1
(1.2.70)
What is the average distance traveled in time t? t
< d(t ) > =
∑
t
< s(t ′ ) > =
t ′ =1
∑ (P+ − P− ) = t (P+ − P− )
(1.2.71)
t ′ =1
This equation justifies defining the mean velocity as v = P+ − P−
(1.2.72)
Since we already have an average displacement it doesn’t make sense to also ask for a typical displacement,as we did with the random walk—the typical displacement is the average one. However, we can ask about the spread of the displacements around the average displacement (t ) 2 = <(d(t)− ) 2 > = −2 < d(t) > 2 + 2 = < d(t )2 > − < d(t) > 2
(1.2.73)
This is called the standard deviation and it reduces to the RMS distance in the unbiased case. For many purposes (t) plays the same role in the biased random walk as in the unbiased random walk. From Eq. (1.2.71) and Eq. (1.2.72) the second term is (vt)2. The first term is:
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2
t t < d(t ) > = < s(t ′) > = < s(t ′)s(t ′′) > t ′=1 t ′ ,t ′′ =1
∑
2
t
=
∑
t ′, t ′′ =1
∑
t ′ ,t ′
+(1 −
t ′ ,t ′ )(P+
2
(1.2.74)
2 + P− − 2P+ P− )
= t + t(t −1)v 2 = t 2v 2 +t (1− v 2 ) Substituting in Eq. (1.2.73): 2
= t(1 − v2 )
(1.2.75)
It is interesting to consider this expression in the two limits v = 1 and v = 0. For v = 1 the walk is deterministic, P+ = 1 and P− = 0, and there is no element of chance; the walker always walks to the right.This is equivalent to the iterative map Eq.(1.1.4).Our result Eq.(1.2.66) is that = 0, as it must be for a deterministic system. However, for smaller velocities,the spreading of the systems increases until at v = 0 we recover the case of the unbiased random walk. The complete probability distribution is given by: P(d ;t ) = P+
(d +t )/ 2
P−
(d−t)/ 2
t (d + t )/2
oddeven t,d
(1.2.76)
For large t the distribution can be found as we did for the unbiased random walk. The work is left to Question 1.2.7.
Q
uestion 1.2.7 Find the long time (continuum) distribution for the biased random walk.
Solution 1.2.7 We use the Sterling approximation as before and take the logarithm of the probability. In addition to the expression from the first line of Eq. (1.2.38) we have an additional factor due to the coefficient of Eq.(1.2.76) which appears in place of the factor of 1/2t. We again define x = d/t, and divide by 2 to allow both odd and even integers. We obtain the expression: ln(P(d,t ))= (t /2)[(1+ x)ln2P+ +(1− x )ln2P− ] −(t /2)[(1 + x)ln(1 + x)+ (1− x)ln(1 − x)]−(1/2)ln(2 t (1− x 2 ))
(1.2.77)
It makes the most sense to expand this around the mean of x, <x> = v. To simplify the notation we can use Eq. (1.2.70) and Eq. (1.2.72) to write: P+ = (1 + v)/2 P− = (1 − v)/2
(1.2.78)
With these substitutions we have: ln(P(d,t ))= (t /2)[(1+ x)ln(1 + v) +(1− x)ln(1 − v)]
(1.2.79)
−(t /2)[(1 + x)ln(1 + x)+ (1− x)ln(1 − x)]−(1/2)ln(2 t (1− x )) 2
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We expand the first two terms in a Taylor expansion around the mean of x and expand the third term inside the logarithm. The first term of Eq.(1.2.79) has only a constant and linear term in a Taylor expansion. These cancel the constant and the first derivative of the Taylor expansion of the second term of Eq. (1.2.79) at x = v. Higher derivatives arise only from the second term: 1 2 ln(P(d,t ))= −(t /2)[ (x − v)2 + (x − v)3 + K] 2 (1− v ) 3(1− v 2 )2 − (1/2)ln(2 t[(1− v 2 ) −2v(x − v) +K]) = −[
(d − vt )2 2 (t)
2
+
(d − vt )3 4
3 (t )
(1.2.80)
+K]−(1/2)ln(2 ( (t )2 − 2v(d − v t) +K))
In the last line we have restored d and used Eq.(1.2.75). Keeping only the first terms in both expansions gives us: 1
P(d ;t ) = 2
(t )2
e −(d−v t)
2
2
2 (t )
(1.2.81)
which is a Gaussian distribution around the mean we obtained before. This implies that aside from the constant velocity, and a slightly modified standard deviation, the distribution remains unchanged. The second term in both expansions in Eq.(1.2.80) become small in the limit of large t, as long as we are not interested in the tail of the distribution. Values of (d − vt) relevant to the main part of the distribution are given by the standard deviation, (t). The second terms in Eq. (1.2.80) are thus reduced by a factor of (t) compared to the first terms in the series. Since (t) grows as the square root of the time, they become insignificant for long times. The convergence is slower, however, than in the unbiased random walk (Questions 1.2.2–1.2.5). ❚
Q
uestion 1.2.8 You are a manager of a casino and are told by the owner that you have a cash flow problem. In order to survive, you have to make sure that nine out of ten working days you have a profit. Assume that the only game in your casino is a roulette wheel. Bets are limited to only red or black with a 2:1 payoff. The roulette wheel has an equal number of red numbers and black numbers and one green number (the house always wins on green). Assume that people make a fixed number of 106 total $1 bets on the roulette wheel in each day. a. What is the maximum number of red numbers on the roulette wheel that will still allow you to achieve your objective? b. With this number of red numbers, how much money do you make on average in each day?
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Solution 1.2.8 The casino wins $1 for every wrong bet and loses $1 for every right bet. The results of bets at the casino are equivalent to a random walk with a bias given by: P+ = (N red +1)/(N red + N black +1)
(1.2.82)
P− = N black /(N red + N black +1)
(1.2.83)
where,as the manager, we consider positive the wins of the casino. The color subscripts can be used interchangeably, since the number of red and black is equal. The velocity of the random walk is given by: v = 1/(2N red + 1)
(1.2.84)
To calculate the probability that the casino will lose on a particular day we must sum the probability that the random walk after 106 steps will result in a negative number. We approximate the sum by an integral over the distribution of Eq. (1.2.81). To avoid problems of notation we replace d with y: 0
∫
1 (t )2
2 =
− vt
1
z0
∫ dy ′e −(y ′)
2
2 (t )
2
−∞
2
2 (t )
(1.2.85)
−∞
∫ dz e −z
−∞
2
∫ dye −(y − t ) v
(t )2
2
−∞
=
0
1
Ploss = dyP(y;t = 106 ) =
2
1 = (1 − erf(z 0 )) 2
z= y′
2 (t )
(1.2.86)
z 0 = −vt / 2 (t )2 = −vt / 2t (1− v 2 ) We have written the probability of loss in a day in terms of the error function erf(x)—the integral of a Gaussian defined by erf(z 0 ) ≡
2
z0
∫ dz e −z
2
(1.2.87)
0
Since erf(∞) = 1
(1.2.88)
we have the expression (1 − erf(z 0 )) ≡
2
∞
∫ dz e −z
2
(1.2.89)
z0
which is also known as the complementary error function erfc(x).
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To obtain the desired constraint on the number of red numbers, or equivalently on the velocity, we invert Eq. (1.2.85) to find a value of v that gives the desired Ploss = 0.1, or erf(z0) = 0.8. Looking up the error function or using iterative guessing on an appropriate computer gives z0 = 0.9062. Inverting Eq. (1.2.86) gives: v=
1 t /2z 0 −1
≈ 2z 0 /t
(1.2.90)
The approximation holds because t is large.The numerical result is v = 0.0013. This g ives us the desired number of each color (inverting Eq. (1.2.84)) of Nred = 371. Of course the result is a very large number and the problem of winning nine out of ten days is a very conservative problem for a casino. Even if we insist on winning ninety-nine out of one hundred days we would have erf(z0) = 0.98, z0 = 1.645, v = 0.0018 and Nred = 275. The profits per day in each case are given by vt, which is approximately $1,300 and $1,800 respectively. Of course this is much less than for bets on a more realistic roulette wheel. Eventually as we reduce the chance of the casino losing and z0 becomes larger, we might become concerned that we are describing the properties of the tail of the distribution when we calculate the fraction of days the casino might lose,and Eq.(1.2.85) will not be very accurate. However, it is not difficult to see that casinos do not have cash flow problems. ❚ In order to generalize the proof of the central limit theorem to the case of a biased random walk, we can treat the continuum case most simply by considering the ˆ where (using d → x for the continuum case): system variable x, xˆ = x − < x > t = x −t < s > = x − vt
(1.2.91)
Only x is a stochastic variable on the right side, v and t are numbers. Since iterations of this variable would satisfy the conditions for the generalized random walk, the generalization of the Gaussian distribution to Eq.(1.2.81) is proved. The discrete case is more difficult to prove because we cannot shift the variable d by arbitrary amounts and continue to consider it as discrete. We can argue the discrete case to be valid on the basis of the result for the continuum case, but a separate proof can be constructed as well.
1.2.4 Master equation approach The Master equation is an alternative approach to stochastic systems,an alternative to Eq. (1.2.5), that is usually applied when time is continuous. We d evelop it starting from the discrete time case. We can rewrite Eq. (1.2.5) in the form of a difference equation for a particular probability P(s). Beginning from: P(s;t) = P(s;t −1) +
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we extract the term where the system remains in the same state: P(s;t) = P(s;t −1) + P(s | s ′)P(s ′ ;t − 1) + P(s |s)P(s;t −1) − P(s;t −1) s ′≠s
∑
(1.2.93)
We use the normalization of probability to write it in terms of the transitions away from this site: P(s;t) = P(s;t −1) + P(s | s ′)P(s ′ ;t − 1) + 1 − P(s ′ | s) P(s;t −1) − P(s;t −1) s ′ ≠s s ′≠s
∑
∑
(1.2.94) Canceling the terms in the bracket that refer only to the probability P(s;t − 1) we write this as a difference equation. On the right appear only the probabilities at different values of the state variable (s′ ≠ s):
∑ ( P(s | s ′)P(s ′;t −1) − P(s ′ | s)P(s;t −1))
P(s,t) − P(s;t −1) =
s ′ ≠s
(1.2.95)
To write the continuum form we reintroduce the time difference between steps ∆t. P(s,t) − P(s;t − ∆t) P(s ′ |s) P(s | s ′) ′ = P(s ;t − ∆t ) − P(s;t − ∆t ) (1.2.96) ∆t ∆t ∆t s ′ ≠s
∑
When the limit of ∆t → 0 is meaningful, it is possible to make the change to the equation P˙ (s,t) =
∑ ( R(s | s ′)P(s ′ ;t ) − R(s ′ |s)P(s;t )) s ′ ≠s
(1.2.97)
Where the ratio P(s | s′)/∆t has been replaced by the rate of transition R(s | s′). Eq.(1.2.97) is called the Master equation and we can consider Eq.(1.2.95) as the discrete time analog. The Master equation has a simple interpretation: The rate of change of the probability of a particular state is the total rate at which probability is being added into that state from all other states,minus the total rate at which probability is leaving the state. Probability is acting like a fluid that is flowing to or from a particular state and is being conserved,as it must be. Eq.(1.2.97) is very much like the continuity equation of fluid flow, where the density of the fluid at a particular place changes according to how much is flowing to that location or from it.We will construct and use the Master equation approach to discuss the problem of relaxation in activated processes in Section 1.4.
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1.3
Thermodynamics and Statistical Mechanics
The field of thermodynamics is easiest to understand in the context of Newtonian mechanics. Newtonian mechanics describes the effect of forces on objects. Thermodynamics describes the effect of heat transfer on objects. When heat is transferred,the temperature of an object changes.Temperature and heat are also intimately related to energy. A hot gas in a piston has a high pressure and it can do mechanical work by applying a force to a piston. By Newtonian mechanics the work is directly related to a transfer of energy. The laws of Newtonian mechanics are simplest to describe using the abstract concept of a point object with mass but no internal structure. The analogous abstraction for thermodynamic laws are materials that are in equilibrium and (even better) are homogeneous. It turns out that even the description of the equilibrium properties of materials is so rich and varied that this is still a primary focus of active research today. Statistical mechanics begins as an effort to explain the laws of thermodynamics by considering the microscopic application of Newton’s laws. Microscopically, the temperature of a gas is found to be related to the kinetic motion of the gas molecules. Heat transfer is the transfer of Newtonian energy from one object to another. The statistical treatment of the many particles of a material, with a key set of assumptions, reveals that thermodynamic laws are a natural consequence of many microscopic particles interacting with each other. Our studies of complex systems will lead us to discuss the properties of systems composed of many interacting parts. The concepts and tools of statistical mechanics will play an important role in these studies, as will the laws of thermodynamics that emerge from them. Thermodynamics also begins to teach us how to think about systems interacting with each other.
1.3.1 Thermodynamics Thermodynamics describes macroscopic pieces of material in equilibrium in terms of macroscopic parameters. Thermodynamics was developed as a result of experience/experiment and,like Newton’s laws,is to be understood as a set of self-consistent definitions and equations. As with Newtonian mechanics, where in its simplest form objects are point particles and friction is ignored,the discussion assumes an idealization that is directly experienced only in special circumstances. However, the fundamental laws, once understood,can be widely applied. The central quantities that are to be defined and related are the energy U, temperature T, entropy S, pressure P, the mass (which we write as the number of particles) N, and volume V. For magnets,the quantities should include the magnetization M, and the magnetic field H. Other macroscopic quantities that are relevant may be added as necessary within the framework developed by thermodynamics.Like Newtonian mechanics,a key aspect of thermodynamics is to understand how systems can be acted upon or can act upon each other. In addition to the quantities that describe the state of a system, there are two quantities that describe actions that may be made on a system to change its state: work and heat transfer.
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The equations that relate the macroscopic quantities are known as the zeroth, first and second laws of thermodynamics. Much of the difficulty in understanding thermodynamics arises from the way the entropy appears as an essential but counterintuitive quantity. It is more easily understood in the context of a statistical treatment included below. A second source of difficulty is that even a seemingly simple material system, such as a piece of metal in a room, is actually quite complicated thermodynamically. Under usual circumstances the metal is not in equilibrium but is emitting a vapor of its own atoms.A thermodynamic treatment of the metal requires consideration not only of the metal but also the vapor and even the air that applies a pressure upon the metal. It is therefore generally simplest to consider the thermodynamics of a gas confined in a closed (and inert) chamb er as a model thermodynamic system. We will discuss this example in detail in Question 1.3.1. The translational motion of the whole system, treated by Newtonian mechanics, is ignored. We begin by defining the concept of equilibrium.A system left in isolation for a long enough time achieves a macroscopic state that does not vary in time.The system in an unchanging state is said to be in equilibrium. Thermodynamics also relies upon a particular type of equilibrium known as thermal equilibrium. Two systems can be brought together in such a way that they interact only by transferring heat from one to the other. The systems are said to be in thermal contact. An example would be two gases separated by a fixed but thermally conducting wall. After a long enough time the system composed of the combination of the two original systems will be in equilibrium. We say that the two systems are in thermal equilibrium with each other. We can generalize the definition of thermal equilibrium to include systems that are not in contact. We say that any two systems are in thermal equilibrium with each other if they do not change their (macroscopic) state when they are brought into thermal contact. Thermal equilibrium does not imply that the system is homogeneous, for example, the two gases may be at different pressures. The zeroth law of thermodynamics states that if two systems are in thermal equilibrium with a third they are in thermal equilibrium with each other. This is not obvious without exp erience with macroscopic objects. The zeroth law implies that the interaction that occurs during thermal contact is not specific to the materials,it is in some sense weak,and it matters not how many or how big are the systems that are in contact. It enables us to define the temperature T as a quantity which is the same for all systems in thermal equilibrium. A more specific definition of the temperature must wait till the second law of thermodynamics. We also define the concept of a thermal reservoir as a very large system such that any system that we are interested in, when brought into contact with the thermal reservoir, will change its state by transferring heat to or from the reservoir until it is in equilibrium with the reservoir, but the transfer of heat will not affect the temperature of the reservoir. Quite basic to the formulation and assumptions of thermodynamics is that the macroscopic state of an isolated system in equilibrium is completely defined by a specification of three parameters: energy, mass and volume (U,N,V). For magnets we must add the magnetization M; we will leave this case for later. The confinement of
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the system to a volume V is understood to result from some form of containment. The state of a system can be characterized by the force per unit area—the pressure P—exerted by the system on the container or by the container on the system, which are the same. Since in equilibrium a system is uniquely described by the three quantities (U,N,V), these determine all the other quantities, such as the pressure P and temperature T. Strictly speaking, temperature and pressure are only defined for a system in equilibrium, while the quantities (U,N,V) have meaning both in and out of equilibrium. It is assumed that for a homogeneous material, changing the size of the system by adding more material in equilibrium at the same pressure and temperature changes the mass, number of particles N, volume V and energy U, in direct proportion to each other. Equivalently, it is assumed that cutting the system into smaller parts results in each subpart retaining the same properties in proportion to each other (see Figs.1.3.1 and 1.3.2). This means that these quantities are additive for different parts of a system whether isolated or in thermal contact or full equilibrium: N=
∑N
V=
∑V
U=
∑U
(1.3.1)
where indexes the parts of the system. This would not be true if the parts of the system were strongly interacting in such a way that the energy depended on the relative location of the parts. Properties such as (U,N,V) that are proportional to the size of the system are called extensive quantities. Intensive quantities are properties that do not change with the size of the system at a given pressure and temperature. The ratio of two extensive quantities is an intensive quantity. Examples are the particle density N/V and the energy density U/V. The assumption of the existence of extensive and intensive quantities is also far from trivial, and corresponds to the intuition that for a macroscopic object,the local properties of the system do not depend on the size of the system. Thus a material may be cut into two parts, or a small part may be separated from a large part, without affecting its local properties. The simplest thermodynamic systems are homogeneous ones,like a gas in an inert container. However we can also use Eq.(1.3.1) for an inhomogeneous system. For example,a sealed container with water inside will reach a state where both water and vapor are in equilibrium with each other. The use of intensive quantities and the proportionality of extensive quantities to each other applies only within a single phase— a single homogeneous part of the system, either water or vapor. However, the additivity of extensive quantities in Eq. (1.3.1) still applies to the whole system. A homogeneous as well as a heterogeneous system may contain different chemical species. In this case the quantity N is replaced by the number of each chemical species Ni and the first line of Eq.(1.3.1) may be replaced by a similar equation for each species.
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T he rmod yn a mics a nd sta t is t i ca l m echa ni cs Figure 1.3.1 Thermodynamics considers macroscopic materials. A basic assumption is that cutting a system into two parts will not affect the local properties of the material and that the energy U, mass (or number of particles) N and the volume V will be divided in the same proportion. The process of separation is assumed to leave the materials under the same conditions of pressure and temperature. ❚
61
Mass N Volume V Energy U
Mass αN Volume αV Energy αU
Mass (1–α)N Volume (1–α)V Energy (1–α)U
Figure 1.3.2 The assumption that the local properties of a system are unaffected by subdivision applies also to the case where a small part of a much larger system is removed. The local properties, both of the small system and of the large system are assumed to remain unchanged. Even though the small system is much smaller than the original system, the small system is understood to be a macroscopic piece of material. Thus it retains the same local properties it had as part of the larger system. ❚
The first law of thermodynamics describes how the energy of a system may change. The energy of an isolated system is conserved. There are two macroscopic processes that can change the energy of a system when the number of particles is fixed.
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The first is work,in the sense of applying a force over a distance, such as driving a piston that compresses a gas. The second is heat transfer. This may be written as: dU = q + w
(1.3.2)
where q is the heat transfer into the system, w is the work done on the system and U is the internal energy of the system. The differential d signifies the incremental change in the quantity U as a result of the incremental process of heat transfer and work. The work performed on a gas (or other system) is the force times the distance applied Fdx, where we write F as the magnitude of the force and dx as an incremental distance. Since the force is the pressure times the area F = PA, the work is equal to the pressure times the volume change or: w = −PAdx = −PdV
(1.3.3)
The negative sign arises because positive work on the system,increasing the system’s energy, occurs when the volume change is negative. Pressure is defined to be positive. If two systems act upon each other, then the energy transferred consists of both the work and heat t ransfer. Each of these are separately equal in magnitude and opposite in sign: dU1 = q21 + w21 dU2 = q12 + w12
(1.3.4)
q12 = −q21 w12 = −w21
where q21 is the heat transfer from system 2 to system 1,and w21 is the work performed by system 2 on system 1. q12 and w12 are similarly defined. The last line of Eq.(1.3.4) follows from Newton’s third law. The other equations follow from setting dU = 0 (Eq. (1.3.2)) for the total system, composed of both of the systems acting upon each other. The second law of thermodynamics given in the following few paragraphs describes a few key aspects of the relationship of the equilibrium state with nonequilibrium states.The statement of the second law is essentially a definition and description of properties of the entropy. Entropy enables us to describe the process of approach to equilibrium. In the natural course of events,any system in isolation will change its state toward equilibrium. A system which is not in equilibrium must therefore undergo an irreversible process leading to equilibrium. The process is irreversible because the reverse process would take us away from equilibrium, which is impossible for a macroscopic system. Reversible change can occur if the state of a system in equilibrium is changed by transfer of heat or by work in such a way (slowly) that it always remains in equilibrium. For every macroscopic state of a system (not necessarily in equilibrium) there exists a quantity S called the entropy of the system. The change in S is positive for any natural process (change toward equilibrium) of an isolated system dS ≥ 0
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For an isolated system, equality holds only in equilibrium when no change occurs. The converse is also true—any possible change that increases S is a natural process. Therefore, for an isolated system S achieves its maximum value for the equilibrium state. The second property of the entropy describes how it is affected by the processes of work and heat transfer during reversible processes. The entropy is affected only by heat transfer and not by work. If we only perform work and do not transfer heat the entropy is constant. Such processes where q = 0 are called adiabatic processes. For adiabatic processes dS = 0. The third property of the entropy is that it is extensive: S=
∑S
(1.3.6)
Since in equilibrium the state of the system is defined by the macroscopic quantities (U,N,V), S is a function of them—S = S(U,N,V)—in equilibrium. The fourth property of the entropy is that if we keep the size of the system constant by fixing both the number of particles N and the volume V, then the change in entropy S with increasing energy U is always positive: S >0 U N ,V
(1.3.7)
where the subscripts denote the (values of the) constant quantities. Because of this we can also invert the function S = S(U,N,V) to obtain the energy U in terms of S, N and V: U = U(S,N,V). Finally, we mention that the zero of the entropy is arbitrary in classical treatments. The zero of entropy does attain significance in statistical treatments that include quantum effects. Having described the properties of the entropy for a single system, we can now reconsider the problem of two interacting systems. Since the entropy describes the process of equilibration, we consider the process by which two systems e quilibrate thermally. According to the zeroth law, when the two systems are in equilibrium they are at the same temperature. The two systems are assumed to be isolated from any other influence,so that together they form an isolated system with energy Ut and entropy St . Each of the subsystems is itself in equilibrium, but they are at different temperatures initially, and therefore heat is t ransferred to achieve equilibrium. The heat transfer is assumed to be performed in a reversible fashion—slowly. The two subsystems are also assumed to have a fixed number of particles N1,N2 and volume V1,V2. No work is done, only heat is transferred. The energies of the two systems U1 and U2 and entropies S1 and S2 are not fixed. The transfer of heat results in a transfer of energy between the two systems according to Eq. (1.3.4), since the total energy Ut = U1 + U2
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is conserved, we have dUt = dU1 + dU2 = 0
(1.3.9)
We will consider the processes of equilibration twice. The first time we will identify the equilibrium condition and the second time we will describe the equilibration. At equilibrium the entropy of the whole system is maximized. Variation of the entropy with respect to any internal parameter will give zero at equilibrium. We can consider the change in the entropy of the system as a function of how much of the energy is allocated to the first system: dSt dS dS = 1 + 2 =0 dU 1 dU 1 dU 1
(1.3.10)
in equilibrium. Since the total energy is fixed, using Eq. (1.3.9) we have: dSt dS dS = 1 − 2 =0 dU 1 dU 1 dU 2
(1.3.11)
dS1 dS 2 = dU 1 dU 2
(1.3.12)
or
in equilibrium. By the definition of the temperature,any function of the derivative of the entropy with respect to energy could be used as the temperature. It is conventional to define the temperature T using: 1 dS = T dU N ,V
(1.3.13)
This definition corresponds to the Kelvin temperature scale.The units of temperature also define the units of the entropy. This definition has the advantage that heat always flows from the system at higher temperature to the system at lower temperature. To prove this last statement, consider a natural small transfer of heat from one system to the other. The transfer must result in the two systems raising their collective entropy: dSt = dS1 + dS2 ≥ 0
(1.3.14)
We rewrite the change in entropy of each system in terms of the change in energy. We recall that N and V are fixed for each of the two systems and the entropy is a function only of the three macroscopic parameters (U,N,V). The change in S for each system may be written as: S dS1 = dU 1 U N 1 ,V 1 (1.3.15) S dS 2 = dU 2 U N 2 ,V 2
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to arrive at: S S dU1 + dU 2 ≥ 0 U N 1 ,V 1 U N 2 ,V 2
(1.3.16)
or using Eq. (1.3.9) and the definition of the temperature (Eq. (1.3.13)) we have: 1 1 − dU1 ≥ 0 T1 T2
(1.3.17)
(T2 −T1) dU1 ≥ 0
(1.3.18)
or: This implies that a natural process of heat transfer results in the energy of the first system increasing (dU1 > 0) if the temperature of the second system is greater than the first ((T2 − T1) > 0), or conversely, ifthe temperature of the second system is less than the temperature of the first. Using the definition of temperature, we can also rewrite the expression for the change in the energy of a system due to heat transfer or work, Eq.(1.3.2). The new expression is restricted to reversible processes. As in Eq. (1.3.2), N is still fixed. Considering only reversible processes means we consider only equilibrium states of the system, so we can write the energy as a function of the entropy U = U(S,N,V). Since a reversible process changes the entropy and volume while keeping this function valid, we can write the change in energy for a reversible process as U U dU = dS + dV S N ,V V N ,S U = TdS + dV V N ,S
(1.3.19)
The first term reflects the effect of a change in entropy and the second reflects the change in volume. The change in entropy is related to heat transfer but not to work. If work is done and no heat is transferred,then the first term is zero. Comparing the second term to Eq. (1.3.2) we find U P = − V N ,S
(1.3.20)
and the incremental change in energy for a reversible process can be written: dU = TdS − PdV
(1.3.21)
This relationship enables us to make direct experimental measurements of entropy changes. The work done on a system, in a reversible or irreversible process, changes the energy of the system by a known amount. This energy can then be extracted in a reversible process in the form of heat. When the system returns to its original state, we
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can quantify the amount of heat transferred as a form of energy. Measured heat transfer can then be related to entropy changes using q = TdS. Our treatment of the fundamentals of thermodynamics was brief and does not contain the many applications necessary for a detailed understanding. The properties of S that we have described are sufficient to provide a systematic treatment of the thermodynamics of macroscopic bodies. However, the entropy is more understandable from a microscopic (statistical) description of matter. In the next section we introduce the statistical treatment that enables contact between a microscopic picture and the macroscopic thermodynamic treatment of matter. We will use it to give microscopic meaning to the entropy and temperature.Once we have developed the microscopic picture we will discuss two applications. The first application, the ideal gas, is discussed in Section 1.3.3. The discussion of the second application,the Ising model of magnetic systems, is postponed to Section 1.6.
1.3.2 The macroscopic state from microscopic statistics In order to develop a microscopic understanding of the macroscopic properties of matter we must begin by restating the nature of the systems that thermodynamics describes. Even when developing a microscopic picture, the thermodynamic assumptions are relied upon as guides. Macroscopic systems are assumed to have an extremely large number N of individual particles (e.g.,at a scale of 1023) in a volume V. Because the size of these systems is so large,they are typically investigated by considering the limit of N →∞ and V → ∞, while the density n = N /V remains constant. This is called the thermodynamic limit. Various properties of the system are separated into extensive and intensive quantities. Extensive quantities are p roportional to the size of the system. Intensive quantities are independent of the size of the system. This reflects the intuition that local properties of a macroscopic object do not depend on the size of the system. As in Figs.1.3.1 and 1.3.2, the system may be cut into two parts, or a small part may be separated from a large part without affecting its local properties. The total energy U of an isolated system in equilibrium, along with the number of particles N and volume V, defines the macroscopic state (macrostate) of an isolated system in equilibrium. Microscopically, the energy of the system E is given in classical mechanics in terms of the complete specification of the individual particle positions, momenta and interaction potentials. Together these define the microscopic state (microstate) of the system. The microstate is defined differently in quantum mechanics but similar considerations apply. When we describe the system microscopically we use the notation E rather than U to describe the energy. The reason for this difference is that macroscopically the energy U has some degree of fuzziness in its definition, though the degree of fuzziness will not enter into our considerations. Moreover, U may also be used to describe the energy of a system that is in thermal equilibrium with another system. However, thinking microscopically, the energy of such a system is not well defined,since thermal contact allows the exchange of energy between the two systems. We should also distinguish between the microscopic and macroscopic concepts of the number of particles and the volume,but since we will not make use of this distinction, we will not do so.
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There are many possible microstates that correspond to a particular macrostate of the system specified only by U,N,V. We now make a key assumption of statistical mechanics—that all of the possible microstates of the system occur with equal probability. The number of these microstates (U,N,V), which by definition depends on the macroscopic parameters, turns out to be central to statistical mechanics and is directly related to the entropy. Thus it determines many of the thermodynamic properties of the system, and can be discussed even though we are not always able to obtain it explicitly. We consider again the problem of interacting systems. As before, we consider two systems (Fig. 1.3.3) that are in equilibrium separately, with state variables (U1,N1,V1) and (U2,N2,V2). The systems have a number of microstates 1(U1,N1,V1) and 2(U2,N2,V2) respectively. It is not necessary that the two systems be formed of the same material or have the same functional form of (U,N,V), so the function is also labeled by the system index. The two systems interact in a limited way, so that they can exchange only energy. The number of particles and volume of each system remains fixed. Conservation of energy requires that the total energy Ut = U1 + U2 remains fixed, but energy may be transferred from one system to the other. As before, our objective is to identify when energy transfer stops and equilibrium is reached. Consider the number of microstates of the whole system t . This number is a function not only of the total energy of the system but also of how the energy is allocated between the systems. So, we write t (U1,U2), and we assume that at any time the energy of each of the two systems is well defined. Moreover, the interaction between the two systems is sufficiently weak so that the number of states of each system
Figure 1.3.3 Illustration of a system formed out of two parts. The text discusses this system when energy is transferred from one part to the other. The transfer of energy on a microscopic scale is equivalent to the transfer of heat on a macroscopic scale, since the two systems are not allowed to change their number of particles or their volume. ❚
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U t , N t ,V t S t (U t , N t ,V t ) t (U t , N t ,V t )
U 1 ,N 1 ,V 1
U 2 ,N 2 ,V 2
S 1 (U 1 ,N 1 ,V 1 )
S 2 (U 2 ,N 2 ,V 2 )
1 (U 1 ,N 1 ,V 1 )
2 (U 2 ,N 2 ,V 2 )
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may be counted independently. Then the total number of microstates is the product of the number of microstates of each of the two systems separately. t(U1,U 2)
=
1(U1)
2(U 2)
(1.3.22)
where we have dropped the arguments N and V, since they are fixed throughout this discussion. When energy is transferred,the number of microstates of each of the two systems is changed. When will the transfer of energy stop? Left on its own,the system will evolve until it reaches the most probable separation of energy. Since any particular state is equally likely, the most probable separation of energy is the separation that gives rise to the greatest possible number of states. When the number of particles is large,the greatest number of states corresponding to a particular energy separation is much larger than the number of states corresponding to any other possible separation. Thus any other possibility is completely negligible. No matter when we look at the system, it will be in a state with the most likely separation of the energ y. For a macroscopic system,it is impossible for a spontaneous transfer of energy to occur that moves the system away from equilibrium. The last paragraph implies that the transfer of energy from one system to the other stops when t reaches its maximum value. Since Ut = U1 + U2 we can find the maximum value of the number of microstates using: t (U 1 ,U t
−U 1)
U1 0=
=0 =
1 (U 1 )
2(U t
U1 (U ) 1 1 2(U 2 )− U1
−U1 ) +
1(U 1 )
2 (U t
−U 1 )
U1
2 (U 2 ) U2
1 (U1 )
(1.3.23)
or 1 (U 1 1)
1 (U 1)
U1
ln
1(U1 )
U1
=
1 (U 2 2)
=
ln
2 (U 2 )
U2
2 (U 2 )
(1.3.24)
U2
The equivalence of these quantities is analogous to the equivalence of the temperature of the two systems in equilibrium. Since the derivatives in the last equation are performed at constant N and V, it appears, by analogy to Eq. (1.3.12), that we can identify the entropy as: S = k ln( (E,N,V)).
(1.3.25)
The constant k, known as the Boltzmann constant, is needed to ensure correspondence of the microscopic counting of states with the macroscopic units of the entropy, as defined by the relationship of Eq. (1.3.13), once the units of temperature and energy are defined. The entropy as defined by Eq.(1.3.25) can be shown to satisfy all of the properties of the thermodynamic entropy in the last section. We have argued that an isolated
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system evolves its macrostate in such a way that it maximizes the number of microstates that correspond to the macrostate. By Eq. (1.3.25), this is the same as the first property of the entropy in Eq. (1.3.5), the maximization of the entropy in equilibrium. Interestingly, demonstrating the second property of the entropy, that it does not change during an adiabatic process, requires further formal developments relating entropy to information that will be discussed in Sections 1.7 and 1.8. We will connect the two discussions and thus be able to demonstrate the second property of the entropy in Chapter 8 (Section 8.3.2). The extensive property of the entropy follows from Eq.(1.3.22). This also means that the number of states at a particular energy grows exponentially with the size of the system. More properly, we can say that experimental observation that the entropy is extensive suggests that the interaction between macroscopic materials, or parts of a single macroscopic material, is such that the microstates of each part of the system may be enumerated independently. The number of microstates can be shown by simple examples to increase with the energy of the system. This corresponds to Eq.(1.3.7). There are also examples where this can be violated, though this will not enter into our discussions. We consider next a second example of interacting systems that enables us to evaluate the meaning of a system in equilibrium with a reservoir at a temperature T. We consider a small part of a much larger system (Fig. 1.3.4). No assumption is necessary regarding the size of the small system; it may be either microscopic or macroscopic. Because of the contact of the small system with the large system, its energy is not
U t ,N t ,V t ,T E({x,p}),N,V U,N,V
Figure 1.3.4 In order to understand temperature we consider a closed system composed o f a large and small system, or equivalently a small system which is part of a much larger system. The larger system serves as a thermal reservoir transferring energy to and from the small system without affecting its own temperature. A microscopic description of this process in terms of a single microscopic state of the small system leads to the Boltzmann probability. An analysis in terms of the macroscopic state of the small system leads to the principle of minimization of the free energy to obtain the equilibrium state of a system at a fixed temperature. This principle replaces the principle of maximization of the entropy, which only applies for a closed system. ❚
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always the same.Energy will be transferred back and forth between the small and large systems. The essential assumption is that the contact between the large and small system does not affect any other aspect of the description of the small system. This means that the small system is in some sense independent of the large system, despite the energy transfer. This is true if the small system is itself macroscopic, but it may also be valid for certain microscopic systems. We also assume that the small system and the large system have fixed numbers of particles and volumes. Our objective is to consider the probability that a particular microstate of the small system will be realized. A microstate is identified by all of the microscopic parameters necessary to completely define this state. We use the notation {x,p} to denote these coordinates. The probability that this particular state will be realized is given by the fraction of states of the whole system for which the small system attains this state. Because there is only one such state for the small system, the probability that this state will be realized is given by (proportional to) a count of the number of states of the rest of the system. Since the large system is macroscopic, we can count this number by using the macroscopic expression for the number of states of the large system: P({x, p}) ∝
R(Ut
− E({x, p}),Nt − N,Vt − V)
(1.3.26)
where E({x,p}),N,V are the energy, number of particles and volume of the microscopic system respectively. E({x,p})is a function of the microscopic parameters {x,p}. Ut ,Nt ,Vt are the energy, number of particles and volume of the whole system,including both the small and large systems. R is the entropy of the large subsystem (reservoir). Since the number of states generally grows faster than linearly as a function of the energy, we use a Taylor expansion of its logarithm (or equivalently a Taylor expansion of the entropy) to find ln
R (U t
= ln
− E({x, p}), N t − N ,Vt −V )
R (U t ,N t
ln − N,V t −V ) +
R (U t ,N t
Et
− N ,Vt −V ) (−E({x ,p})) N t ,V t
1 (−E({x ,p})) (1.3.27) kT where we have not expanded in the number of particles and the volume because they are unchanging. We take only the first term in the expansion, because the size of the small system is assumed to be much smaller than the size of the whole system. Exponentiating gives the relative probability of this particular microscopic state. = ln
R (U t ,N t
R(Ut
− N,V t −V ) +
− E({x,p}),Nt − N,Vt − V) =
R(Ut ,Nt
− N,Vt − V)e−E({x,p})/kT (1.3.28)
The probability of this particular state must be normalized so that the sum over all states is one.Since we are normalizing the probability anyway, the constant coefficient does not affect the result. This gives us the Boltzmann probability distribution:
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P({x, p}) = Z=
1 −E({x ,p })/ kT e Z
∑ e −E({ x,p }) /kT
71
(1.3.29)
{x,p}
Eq. (1.3.29) is independent of the states of the large system and depends only on the microscopic description of the states of the small system. It is this expression which generally provides the most convenient starting point for a connection between the microscopic description of a system and macroscopic thermodynamics. It identifies the probability that a particular microscopic state will be realized when the system has a well-defined temperature T. In this way it also provides a microscopic meaning to the macroscopic temperature T. It is emphasized that Eq.(1.3.29) describes both microscopic and macroscopic systems in equilibrium at a temperature T. The probability of occurrence of a particular state should be related to the description of a system in terms of an ensemble. We have found by Eq. (1.3.29) that a system in thermal equilibrium at a temperature T is represented by an ensemble that is formed by taking each of the states in proportion to its Boltzmann probability. This ensemble is known as the canonical ensemble. The canonical ensemble should be contrasted with the assumption that each state has equal probability for isolated systems at a particular energy. The ensemble of fixed energy and equal a priori probability is known as the microcanonical ensemble. The canonical ensemble is both easier to discuss analytically and easier to connect with the physical world. It will be generally assumed in what follows. We can use the Boltzmann probability and the definition of the canonical ensemble to obtain all of the thermodynamic quantities. The macroscopic energy is given by the average over the microscopic energy using: U=
1 E({x , p})e −E({x ,p })/ kT Z {x,p}
∑
(1.3.30)
For a macroscopic system,the average value of the energy will always be observed in any specific measurement, despite the Boltzmann probability that allows all energies. This is because the number of states of the system rises rapidly with the energy. This rapid growth and the exponential decrease of the probability with the energy results in a sharp peak in the probability distribution as a function of energy. The sharp peak in the probability distribution means that the probability of any other energy is negligible. This is discussed below in Question 1.3.1. For an isolated macroscopic system, we were able to identify the equilibrium state from among other states of the system using the principle of the maximization of the entropy. There is a similar procedure for a macroscopic system in contact with a thermal reservoir at a fixed temperature T. The important point to recognize is that when we had a closed system,the energy was fixed. Now, however, the objective becomes to identify the energy at equilibrium. Of course, the energy is given by the average in
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Eq.(1.3.30). However, to generalize the concept of maximizing the entropy, it is simplest to reconsider the problem of the system in contact with the reservoir when the small system is also macroscopic. Instead of considering the probability of a particular microstate of well-defined energy E, we consider the probability of a macroscopic state of the system with an energy U. In this case, we find the equilibrium state of the system by maximizing the number of states of the whole system, or alternatively of the entropy: ln (U, N,V ) + ln R (Ut −U,N t − N ,Vt −V) = S(U,N ,V )/k + SR (U t −U ,N t − N ,Vt −V )/k 1 = S(U,N ,V )/k + SR (U t ,N t − N,V t −V )/k + (−U) kT
(1.3.31)
To find the equilibrium state,we must maximize this expression for the entropy of the whole system. We can again ignore the constant second term. This leaves us with quantities that are only char acterizing the small system we are interested in, and the temperature of the reservoir. Thus we can find the equilibrium state by maximizing the quantity S − U/T
(1.3.32)
It is conventional to rewrite this and, rather than maximizing the function in Eq. (1.3.32), to minimize the function known as the free energy: F = U − TS
(1.3.33)
This suggests a simple physical significance of the process of change toward equilibrium. At a fixed temperature, the system seeks to minimize its energy and maximize its entropy at the same time. The relative importance of the entropy compared to the energy is set by the temperature. For high temperature, the entropy becomes more dominant, and the energy rises in order to increase the entropy. At low temperature, the energy becomes more dominant, and the energy is lowered at the expense of the entropy. This is the precise statement of the observation that “everything flows downhill.” The energy entropy competition is a balance that is rightly considered as one of the most basic of physical phenomena. We can obtain a microscopic expression for the free energy by an exercise that begins from a microscopic expression for the entropy: S = k ln( ) = k ln {x,p}
∑ E ({ x,p } ),U
(1.3.34)
The summation is over all microscopic states. The delta function is 1 only when E({x, p}) = U. Thus the sum counts all of the microscopic states with energy U. Strictly speaking, the function is assumed to be slightly “fuzzy,” so that it gives 1 when E({x,p}) differs from U by a small amount on a macroscopic scale, but by a large amount in terms of the differences between energies of microstates. We can then write
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S = k ln( ) = k ln {x ,p }
∑
=
U + k ln T {x,p}
∑
E ({ x,p } ),U e
E({ x ,p} ),U e
−E ( {x,p }) / kT U /kT
e
73
(1.3.35)
−E( {x ,p }) / kT
Let us compare the sum in the logarithm with the expression for Z in Eq.(1.3.29). We will argue that they are the same. This discussion hinges on the rapid increase in the number of states as the energy increases. Because of this rapid growth,the value of Z in Eq.(1.3.29) actually comes from only a narrow region of energy. We know from the expression for the energy average, Eq.(1.3.30),that this narrow region of energy must be at the energy U. This implies that for all intents and purposes the quantity in the brackets of Eq. (1.3.35) is equivalent to Z. This argument leads to the expression: U + k ln Z T
(1.3.36)
F = −kTlnZ
(1.3.37)
S= Comparing with Eq. (1.3.33) we have
Since the Boltzmann probability is a convenient starting point,this expression for the free energy is often simpler to evaluate than the expression for the entropy, Eq. (1.3.34).A calculation of the free energy using Eq.(1.3.37) provides contact between microscopic models and the macroscopic behavior of thermodynamic systems. The Boltzmann normalization Z, which is directly related to the free energy is also known as the partition function. We can obtain other thermodynamic quantities directly from the free energy. For example, we rewrite the expression for the energy Eq. (1.3.30) as: U=
1 E({x , p})e − Z {x ,p}
∑
E({x,p})
=−
ln(Z )
=
F
(1.3.38)
where we use the notation = 1/ kT. The entropy can be obtained using this expression for the energy and Eq. (1.3.33) or (1.3.36).
Q
uestion 1.3.1 Consider the possibility that the macroscopic energy of a system in contact with a thermal reservoir will deviate from its typical value U. To do this expand the probability distribution of macroscopic energies of a system in contact with a reservoir around this value. How large are the deviations that occur? Solution 1.3.1 We considered Eq.(1.3.31) in order to optimize the entropy and find the typical value of the energy U. We now consider it again to find the distribution of probabilities of values of the energy around the value U similar to the way we discussed the distribution of microscopic states {x, p} in Eq.(1.3.27). To do this we distinguish between the observed value of the
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energy U′ and U. Note that we consider U′ to be a macroscopic energy, though the same derivation could be used to obtain the distribution of microscopic energies. The probability of U′ is given by: P(U ′) ∝ (U ′, N,V )
R (U t
′ )/k +S R (U t −U ′ )/ k
−U ′ ,N t − N,Vt −V ) = e S (U
(1.3.39)
In the latter form we ignore the fixed arguments N and V. We expand the logarithm of this expression around the expected value of energy U: S(U ′) +SR (U t −U ′) = S(U )/k + SR (Ut −U)/k +
1 d 2S(U) 1 d 2S(U t −U) 2 (U − U ′ ) + (U −U ′ )2 2 2 2k dU 2k dU t (1.3.40)
where we have kept terms to second order. The first-order terms, which are of the form (1/kT)(U′ − U), have opposite signs and therefore cancel. This implies that the probability is a maximum at the expected energy U. The second derivative of the entropy can be evaluated using: d 2S(U) dU
2
=
d 1 1 1 1 =− 2 =− 2 dU T dU /dT T T CV
(1.3.41)
where CV is known as the specific heat at constant volume.For our purposes, its only relevant property is that it is an extensive quantity. We can obtain a similar expression for the reservoir and define the reservoir specific heat CVR. Thus the probability is: P(U ′) ∝e −(1/ 2kT
2
)(1 /C V +1/ CVR )(U −U ′ )
2
≈ e −(1/ 2kT
2
)(1 /C V )(U −U ′)
2
(1.3.42)
where we have left out the (constant) terms that do not depend on U′. Because CV and CVR are extensive quantities and the reservoir is much bigger than the small system, we can neglect 1/CVR compared to 1/CV. The result is a Gaussian distribution (Eq. (1.2.39)) with a standard deviation = T√kCV
(1.3.43)
This describes the characteristic deviation of the energy U′ from the average or typical energy U. However, since CV is extensive, the square root means that the deviation is proportional to √N. Note that the result is consistent with a random walk of N steps. So for a large system of N ∼ 1023 particles,the possible deviation in the energy is smaller than the energy by a factor of (we are neglecting everything but the N dependence) 1012—i.e., it is undetectable. Thus the energy of a thermodynamic system is very well defined. ❚
1.3.3 Kinetic theory of gases and pressure In the previous section, we described the microscopic analog of temperature and entropy. We assumed that the microscopic analog of energy was understood,and we de-
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veloped the concept of free energy and its microscopic analog. One quantity that we have not discussed microscopically is the pressure. Pressure is a Newtonian concept— the force per unit area. For various reasons,it is helpful for us to consider the microscopic origin of pressure for the example of a simplified model of a gas called an ideal gas. In Question 1.3.2 we use the ideal gas as an example of the thermodynamic and statistical analysis of materials. An ideal gas is composed of indistinguishable point particles with a mass m but with no internal structure or size. The interaction between the particles is neglected, so that the energy is just their kinetic energy. The particles do interact with the walls of the container in which the gas is confined. This interaction is simply that of reflection—when the par ticle is incident on a wall, the component of its velocity perpendicular to the wall is reversed.Energy is conserved. This is in accordance with the expectation from Newton’s laws for the collision of a small mass with a much larger mass object. To obtain an expression for the pressure, we must suffer with some notational hazards,as the pressure P, probability of a particular velocity P(v) and momentum of a particular particle pi are all designated by the letter P but with different case, arguments or subscripts.A bold letter F is used briefly for the force,and otherwise F is used for the free energy. We rely largely upon context to distinguish them. Since the objective of using an established notation is to make contact with known concepts,this situation is sometimes preferable to introducing a new notation. Because of the absence of collisions between different particles of the gas, there is no communication between them, and each of the particles bounces around the container on its own course. The pressure on the container walls is given by the force per unit area exerted on the walls,as illustrated in Fig. 1.3.5. The force is given by the action of the wall on the gas that is needed to reverse the momenta of the incident particles between t and t + ∆t : P=
F 1 = A A∆t
∑ m∆v i
(1.3.44)
i
where |F | is the magnitude of the force on the wall. The latter expression relates the pressure to the change in the momenta of incident particles per unit area of the wall. A is a small but still macroscopic area,so that this part of the wall is flat. Microscopic roughness of the surface is neglected. The change in velocity ∆vi of the particles during the time ∆t is zero for particles that are not incident on the wall. Particles that hit the wall between t and t + ∆t are moving in the direction of the wall at time t and are near enough to the wall to reach it during ∆t. Faster particles can reach the wall from farther away, but only the velocity perpendicular to the wall matters. Denoting this velocity component as v⊥, the maximum distance is v⊥∆t (see Fig. 1.3.5). If the particles have velocity only perpendicular to the wall and no velocity parallel to the wall,then we could count the incident particles as those in a volume Av⊥∆t. We can use the same expression even when particles have a velocity parallel to the surface, because the parallel velocity takes particles out of and into this volume equally.
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Figure 1.3.5 Illustration of a gas of ideal particles in a container near one of the walls. Particles incident on the wall are reflected, reversing their velocity perpendicular to the wall, and not affecting the other components of their velocity. The wall experiences a pressure due to the collisions and applies the same pressure to the gas. To calculate the pressure we must count the number of particles in a unit of time ∆t with a particular perpendicular velocity v⊥ that hit an area A. This is equivalent to counting the number of particles with the velocity v⊥ in the box shown with one of its sides of length ∆tv⊥. Particles with velocity v⊥ will hit the wall if and only if they are in the box. The same volume of particles applies if the particles also have a velocity parallel to the surface, since this just skews the box, as shown, leaving its height and base area the same. ❚
Another way to say this is that for a particular parallel velocity we count the particles in a sheared box with the same height and base and therefore the same volume. The total number of particles in the volume,(N / V)Av⊥∆t, is the volume times the density (N /V). Within the volume Av⊥∆t, the number of particles that have the velocity v⊥ is given by the number of particles in this volume times the probability P(v⊥) that a particle has its perpendicular velocity component equal to v⊥. Thus the number of par-
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ticles incident on the wall with a particular velocity perpendicular to the wall v⊥ is given by N AP(v ⊥ )v⊥ ∆t V
(1.3.45)
The total change in momentum is found by multiplying this by the change in momentum of a single particle reflected by the collision, 2mv⊥, and integrating over all velocities. ∞
1 m∆v i = NA∆t dv⊥ P( v⊥ )v⊥ (2m v ⊥ ) V 0
∑ i
∫
(1.3.46)
Divide this by A∆t to obtain the change in momentum per unit time per unit area, which is the pressure (Eq. (1.3.44)), ∞
1 P = N dv⊥ P(v⊥ )v⊥ (2mv ⊥) V 0
∫
(1.3.47)
We rewrite this in terms of the average squared velocity perpendicular to the surface ∞
P=
∞
2 2 2 N N N m 2 dv⊥ P(v⊥ )v ⊥ = m dv ⊥ P(v ⊥)v ⊥ = m < v⊥ > V V −∞ V 0
∫
∫
(1.3.48)
where the equal probability of having positive and negative velocities enables us to extend the integral to −∞ while eliminating the factor of two. We can rewrite Eq.(1.3.48) in terms of the average square magnitude of the total velocity. There are three components of the velocity (two parallel to the surface). The squares of the velocity components add to give the total velocity squared and the averages are equal: < v2 > = < v⊥2 + v22 + v32 > = 3 < v⊥ 2 >
(1.3.49)
where v is the magnitude of the particle velocity. The pressure is: P=
N 1 m < v2 > V 3
(1.3.50)
Note that the wall does not influence the probability of having a particular velocity nearby. Eq. (1.3.50) is a microscopic expression for the pressure, which we can calculate using the Boltzmann probability from Eq. (1.3.29). We do this as part of Question 1.3.2.
Q
uestion 1.3.2 Develop the statistical description of the ideal gas by obtaining expressions for the thermodynamic quantities Z, F, U, S and P, in terms of N, V, and T. For hints read the first three paragraphs of the solution.
Solution 1.3.2 The primary task of statistics is counting. To treat the ideal gas we must count the number of microscopic states to obtain the entropy,
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or sum over the Boltzmann probability to obtain Z and the free energy. The ideal gas presents us with two difficulties. The first is that each particle has a continuum of possible locations. The second is that we must treat the particles as microscopically indistinguishable. To solve the first problem, we have to set some interval of position at which we will call a particle here different from a particle there. Moreover, since a particle at any location may have many different velocities, we must also choose a difference of velocities that will be considered as distinct.We define the interval of position to be ∆x and the interval of momentum to be ∆p. In each spatial dimension,the positions between x and x +∆x correspond to a single state,and the momenta between p and p + ∆p correspond to a single state. Thus we consider as one state of the system a particle which has position and momenta in a six-dimensional box of a size ∆x3∆p3. The size of this box enters only as a constant in classical statistical mechanics, and we will not be concerned with its value. Quantum mechanics identifies it with ∆x3∆p3 = h3, where h is Planck’s constant, and for convenience we adopt this notation for the unit volume for counting. There is a subtle but important choice that we have made. We have chosen to make the counting intervals have a fixed width ∆p in the momentum. From classical mechanics,it is not entirely clear that we should make the intervals of fixed width in the momentum or, for example,make them fixed in the energy ∆E. In the latter case we would count a single state between E and E +∆E. Since the energy is proportional to the square of the momentum,this would give a different counting. Quantum mechanics provides an unambiguous answer that the momentum intervals are fixed. To solve the problem of the indistinguishability of the particles, we must remember every time we count the number of states of the system to divide by the number of possible ways there are to interchange the particles, which is N!. The energy of the ideal gas is given by the kinetic energy of all of the particles: N
2
1 2 N pi E({x, p}) = m vi = 2 2m i =1 i =1
∑
∑
(1.3.51)
where the velocity and momentum of a particle are three-dimensional vectors with magnitude vi and pi respectively. We start by calculating the partition function (Boltzmann normalization) Z from Eq. (1.3.29) N
pi
2
N
pi
2
−∑ −∑ 1 1 Z= e i =12mkT = e i =1 2mkT N! {x ,p} N!
∑
∫
N
d 3x i d 3 pi
i =1
h3
∏
(1.3.52)
where the integral is to be evaluated over all possible locations of each of the N particles of the system. We have also included the correction to over-
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counting, N!. Since the particles do not see each other, the energy is a sum over each particle energy. The integrals separate and we have: p2 − 1 1 3 3 2mkT Z= e d xd p N! h 3
N
∫
(1.3.53)
The position integral gives the volume V, immediately giving the dependence of Z on this macroscopic quantity. The integral over momentum can be evaluated giving:
∫
e
−
2
p 2mkT d 3 p
∞
=4
∫
2
p dpe
0
= 4 (2mkT)3/ 2 − a
−
2
p 2mkT
∞
∫
= 4 (2mkT)3/ 2 y 2dye − y
2
0
∞
2 −ay = 4 (2mkT)3/ 2 − dy e a a =1 0
∫
3/ 2
= (2 mkT)
1 a =1 2
a (1.3.54)
and we have that Z (V ,T,N) =
3N VN 2 2 mkT /h N!
2
(1.3.55)
We could have simplified the integration by recognizing that each component of the momentum px,py and pz can be integrated separately, giving 3N independent one-dimensional integrals and leading more succinctly to the result. The result can also be written in terms of a natural length (T) that depends on temperature (and mass): (T) = (h2 / 2 mkT )1/2 Z (V ,T,N) =
VN N! (T)3 N
(1.3.56) (1.3.57)
From the partition function we obtain the free energy, making use of Sterling’s approximation (Eq. (1.2.36)): F = kTN(lnN − 1) − kTN ln(V / (T )3)
(1.3.58)
where we have neglected terms that grow less than linearly with N. Terms that vary as ln(N) vanish on a macroscopic scale. In this form it might appear that we have a problem,since the N ln(N) term from Sterling’s approximation to the factorial does not scale proportional to the size of the system, and F is an extensive quantity. However, we must also note the N ln(V) term, which we can combine with the N ln(N) term so that the extensive nature is apparent: F = kTN[lnN (T)3/V) − 1]
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It is interesting that the factor of N!,and thus the indistinguishability of particles,is necessary for the free energy to be extensive. If the particles were distinguishable,then cutting the system in two would result in a different counting, since we would lose the states corresponding to particles switching from one part to the other. If we combined the two systems back together, there would be an effect due to the mixing of the distinguishable particles (Question 1.3.3). The energy may be obtained from Eq. (1.3.38) (any of the forms) as: 3 U = NkT 2
(1.3.60)
which provides an example of the equipartition the orem, which says that each degree of freedom (position-momentum pair) of the system carries kT / 2 of energy in equilibrium.Each of the three spatial coordinates of each particle is one degree of freedom. The expression for the entropy (S = (U − F)/T) S = kN[ln(V/N (T)3) + 5/2]
(1.3.61)
shows that the entropy per particle S/N grows logarithmically with the volume per particleV /N. Using the expression for U, it may be written in a form S(U,N,V). Finally, the pressure may be obtained from Eq.(1.3.20), but we must be careful to keep N and S constant rather than T. We have P =−
U V
N ,S
3 T = − Nk 2 V
(1.3.62) N ,S
Taking the same derivative of the entropy Eq. (1.3.61) gives us (the derivative of S with S fixed is zero): 1 3 T 0= − − V 2 V
(1.3.63) N ,S
Substituting, we obtain the ideal gas equation of state: PV = NkT
(1.3.64)
which we can also obtain from the microscopic expression for the pressure— Eq.(1.3.50). We describe two ways to do this.One way to obtain the pressure from the microscopic expression is to evaluate first the average of the energy N
U = < E({x ,p}) > =
1
1
∑ 2m < vi2 > = N 2 m < v 2 >
(1.3.65)
i=1
This may be substituted in to Eq. (1.3.60) to obtain
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1 3 m < v 2 > = kT 2 2
(1.3.66)
which may be substituted directly in to Eq. (1.3.50). Another way is to obtain the average squared velocity directly. In averaging the velocity, it doesn’t matter which particle we choose. We choose the first particle: N
< v12 > = 3 < v12⊥ > =
pi
2
−∑ 1 3v12⊥ e i =12mkT N!
∫
N
pi
2
−∑ 1 e i =1 2mkT N!
∫
N
∏ i=1
N
∏ i=1
d3 x i d 3 p i h3
d 3 x id 3 p i
(1.3.67)
h3
where we have further chosen to average over only one of the components of the velocity of this particle and multiply by three. The denominator is the normalization constant Z. Note that the factor 1/N!, due to the indistinguishability of particles, appears in the numerator in any ensemble average as well as in the denominator, and cancels. It does not affect the Boltzmann probability when issues of distinguishability are not involved. There are 6N integrals in the numerator and in the denominator of Eq. (1.3.67). All integrals factor into one-dimensional integrals.Each integral in the numerator is the same as the corresponding one in the denominator, except for the one that involves the particular component of the velocity we are interested in. We cancel all other integrals and obtain:
< v12
> =3
< v12⊥
∫ > =3
2 v1⊥ e
∫e
−
−
2
p1 ⊥ 2mkT dp
1⊥
2
p1⊥ 2mkT dp
1⊥
2
∫
2 −y 2kT y e dy 2kT 1 = 3( ) = 3( )( ) −y2 m m 2 e dy
∫
(1.3.68) The integral is performed by the same technique as used in Eq.(1.3.54). The result is the same as by the other methods. ❚
Q
uestion 1.3.3 An insulated box is divided into two compartments by a partition. The two compartments contain two different ideal gases at the same pressure P and temperature T. The first gas has N1 particles and the second has N2 particles. The partition is punctured. Calculate the resulting change in thermodynamic parameters (N, V, U, P, S, T, F). What changes in the analysis if the two gases are the same, i.e., if they are composed of the same type of molecules? Solution 1.3.3 By additivity the extrinsic properties of the whole system before the puncture are (Eq. (1.3.59)–Eq. (1.3.61)):
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3 U 0 =U 1 +U 2 = (N 1 + N 2 )kT 2 V 0 = V1 +V 2 S0 = kN1 [ln(V1 /N 1 (T)3 ) + 5/2] +kN 2[ln(V 2 / N 2 (T)3 ) + 5/2]
(1.3.69)
F0 = kTN1[ln(N1 (T)3 /V1 )− 1]+kTN 2[ln(N 2 (T)3 /V2 ) −1] The pressure is intrinsic, so before the puncture it is (Eq. (1.3.64)): P0 = N1kT/V1 = N2kT/V2
(1.3.70)
After the puncture, the total energy remains the same, because the whole system is isolated. Because the two gases do not interact with each other even when they are mixed, their properties continue to add after the puncture. However, each gas now occupies the whole volume, V1 + V2. The expression for the energy as a function of temperature remains the same,so the temperature is also unchanged. The pressure in the container is now additive: it is the sum of the pressure of each of the gases: P = N1kT/(V1 + V2) + N2kT/(V1 + V2) = P0
(1.3.71)
i.e., the pressure is unchanged as well. The only changes are in the entropy and the free energy. Because the two gases do not interact with each other, as with other quantities, we can write the total entropy as a sum over the entropy of each gas separately: S = kN1[ln((V1 + V2)/N1 (T)3) + 5/2] + kN2[ln((V1 + V2)/N2 (T)3) + 5/2]
(1.3.72)
= S0 + (N1 + N2)k ln(V1 + V2) − N1k ln(V1) − N2k ln(V2) If we simplify to the case V1 = V2, we have S = S0 + (N1 + N2)k ln(2). Since the energy is unchanged, by the relationship of free energy and entropy (Eq. (1.3.33)) we have: F = F0 − T(S − S0)
(1.3.73)
If the two gases are composed of the same molecule,there is no change in thermodynamic parameters as a result of a puncture. Mathematically, the difference is that we replace Eq. (1.3.72) with: S = k(N1 + N2)[ln((V1 + V2)/(N1 + N2) (T)3) + 5/2] = S0 (1.3.74) where this is equal to the original entropy because of the relationship N1/V1 = N2 / V2 from Eq. (1.3.70). This example illustrates the effect of indistinguishability. The entropy increases after the puncture when the gases are different, but not when they are the same. ❚
Q
uestion 1.3.4 An ideal gas is in one compartment of a two-compartment sealed and thermally insulated box. The compartment it is in has a volume V1. It has an energy U0 and a number of particles N0. The second com-
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partment has volume V2 and is empty. Write expressions for the changes in all thermodynamic parameters (N, V, U, P, S, T, F) if a. the barrier between the two compartments is punctured and the gas expands to fill the box. b. the barrier is moved slowly, like a piston, expanding the gas to fill the box. Solution 1.3.4 Recognizing what is conserved simplifies the solution of this type of problem. a. The energy U and the number of particles N are conserved. Since the volume change is given to us explicitly, the expressions for T (Eq. (1.3.60)), F (Eq. (1.3.59)), S (Eq. (1.3.61)), and P (Eq. (1.3.64)) in terms of these quantities can be used. N = N0 U = U0 V = V1 + V2
(1.3.75)
T = T0 3
F = kTN[ln(N (T) /(V1 + V2)) − 1] = F0 + kTN ln(V1 + V2)) S = kN[ln((V1 + V2) /N T)3) + 5/2] = S0 + kN ln((V1 + V2)/V1) P = NkT/ V = NkT/(V1 + V2) = P0V1 /(V1 + V2) b. The process is reversible and no heat is transferred,thus it is adiabatic— the entropy is conserved. The number of particles is also conserved: N = N0
(1.3.76)
S = S0
Our main task is to calculate the effect of the work done by the gas pressure on the piston. This causes the energy of the gas to decrease,and the temperature decreases as well. One way to find the change in temperature is to use the conservation of entropy, and Eq. (1.3.61), to obtain that V/ (T)3 is a constant and therefore: T ∝ V-2/3
(1.3.77)
Thus the temperature is given by: V +V T =T0 1 2 V1
−2 / 3
(1.3.78)
Since the temperature and energy are proportional to each other (Eq. (1.3.60)), similarly: V +V 2 U = U 0 1 V1
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(1.3.79)
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The free-energy expression in Eq. (1.3.59) changes only through the temperature prefactor: F = kTN[ln(N (T)3 /V ) −1]= F0
V +V 2 T = F0 1 T0 V1
−2 / 3
(1.3.80)
Finally, the pressure (Eq. (1.6.64)): V +V 2 TV P = NkT /V = P0 0 = P0 1 T0V V1
−5 / 3
(1.3.81) ❚
The ideal gas illustrates the significance of the Boltzmann distribution. Consider a single particle. We can treat it either as part of the large system or as a subsystem in its own right. In the ideal gas, without any interactions, its energy would not change. Thus the particle would not be described by the Boltzmann probability in Eq. (1.3.29). However, we can allow the ideal gas model to include a weak or infrequent interaction (collision) between particles which changes the particle’s energy. Over a long time compared to the time between collisions, the particle will explore all possible positions in space and all possible momenta. The probability of its being at a particular position and momentum (in a region d 3xd 3p) is given by the Boltzmann distribution: e
−
∫e
p2 2mkT d 3 pd 3 x /h3
−
(1.3.82)
2
p 2mkT d 3 pd 3 x /h 3
Instead of considering the trajectory of this particular particle and the effects of the (unspecified) collisions, we can think of an ensemble that represents this particular particle in contact with a thermal reservoir. The ensemble would be composed of many different particles in different boxes. There is no need to have more than one particle in the system. We do need to have some mechanism for energy to be transferred to and from the particle instead of collisions with other particles. This could happen as a result of the collisions with the walls of the box if the vibrations of the walls give energy to the particle or absorb energy from the particle. If the wall is at the temperature T, this would also give rise to the same Boltzmann distribution for the particle. The probability of a particular particle in a particular box being in a particular location with a particular momentum would be given by the same Boltzmann probability. Using the Boltzmann probability distribution for the velocity, we could calculate the average velocity of the particle as:
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∫v e 2
< v 2 >= 3 < v ⊥2 >= 3
⊥
∫
e
−
−
p
2
2mkT d
85
2
3
3
3
pd x /h
2
p 2mkT d 3 pd 3x
/h 3
=
∫
2 v ⊥e
∫
e
−
−
p⊥ 2mkT dp p⊥
2
2mkT dp
⊥
=
3kT (1.3.83) m
⊥
which is the same result as we obtained for the ideal gas in the last part of Question 1.3.2. We could even consider one coordinate of one particle as a separate system and arrive at the same conclusion.Our description of systems is actually a description of coordinates. There are differences when we consider the particle to be a member of an e nsemble and as one par ticle of a gas. In the ensemble, we do not need to consider the distinguishability of particles. This does not affect any of the properties of a single particle. This discussion shows that the ideal gas model may be viewed as quite close to the basic concept of an ensemble.Gene ralize the physical particle in three dimensions to a point with coordinates that describe a complete system. These coordinates change in time as the system evolves according to the rules of its dynamics.The ensemble represents this system in the same way as the ideal gas is the ensemble of the particle. The lack of interaction between the different members of the ensemble,and the existence of a transfer of energy to and from each of the systems to generate the Boltzmann probability, is the same in each of the cases. This analogy is helpful when thinking about the nature of the ensemble.
1.3.4 Phase transitions—first and second order In the previous section we constructed some of the underpinnings of thermodynamics and their connection with microscopic descriptions of materials using statistical mechanics. One of the central conclusions was that by minimizing the free energy we can find the equilibrium state of a material that has a fixed number of particles, volume and temperature. Once the free energy is minimized to obtain the equilibrium state of the material, the energy, entropy and pressure are uniquely determined. The free energy is also a function of the temperature, the volume and the number of particles. One of the important properties of materials is that they can change their properties suddenly when the temperature is changed by a small amount. Examples of this are the transition of a solid to a liquid, or a liquid to a gas. Such a change is known as a phase transition. Each well-defined state of the material is considered a particular phase. Let us consider the process of minimizing the free energy as we vary the temperature. Each of the properties of the material will, in general, change smoothly as the temperature is varied. However, special circumstances might occur when the minimization of the free energy at one temperature results in a very different set of
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properties of the material from this minimization at a slightly different temperature. This is illustrated in a series of frames in Fig. 1.3.6, where a schematic of the free energy as a function of some macroscopic parameter is illustrated. The temperature at which the jump in properties of the material occurs is called the critical or transition temperature,Tc . In general,all of the properties of the material except for the free energy jump discontinuously at Tc . This kind of phase transition is known as a first-order phase transition. Some of the properties of a first-order phase transition are that the two phases can coexist at the transition temperature so that part of the material is in one phase and part in the other. An example is ice floating in water. If we start from a temperature below the transition temperature—with ice—and add heat to the system gradually, the temperature will rise until we reach the transition temperature. Then the temperature will stay fixed as the material converts from one phase to the other—from ice to water. Once the whole system is converted to the higher temperature phase, the temperature will start to increase again.
Figure 1.3.6 Each of the curves represents the variation of the free energy of a system as a function of macroscopic parameters. The different curves are for different temperatures. As the temperature is varied the minimum of the free energy all of a sudden switches from one set of macroscopic parameters to another. This is a first-order phase transition like the melting of ice to form water, or the boiling of water to form steam. Below the ice-to-water phase transition the macroscopic parameters that describe ice are the minimum of the free energy, while above the phase transition the macroscopic parameters that describe water are the minimum of the free energy. ❚
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T c +2∆T
T c +∆T
Tc
T c –∆T
T c –2∆T
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The temperature Tc at which a transition occurs depends on the number of particles and the volume of the system. Alternatively, it may be considered a function of the pressure. We can draw a phase-transition diagram (Fig. 1.3.7) that shows the transition temperature as a function of pressure. Each region of such a diagram corresponds to a particular phase. There is another kind of phase transition, known as a second-order phase transition, where the energy and the pressure do not change discontinuously at the phasetransition point. Instead, they change continuously, but they are nonanalytic at the transition temperature.A common way that this can occur is illustrated in Fig. 1.3.8. In this case the single minimum of the free energy breaks into two minima as a function of temperature. The temperature at which the two minima appear is the transition temperature. Such a second-order transition is often coupled to the existence of first-order transitions. Below the second-order transition temperature, when the two minima exist, the variation of the pressure can change the relative energy of the two minima and cause a first-order transition to occur. The first-order transition occurs at a particular pressure Pc(T) for each temperature below the second-order transition temperature. This gives rise to a line of first-order phase transitions. Above the second-order transition temperature, there is only one minimum, so that there are
P water
2nd order
ice
1st order
steam
T Figure 1.3.7 Schematic phase diagram of H2O showing three phases — ice, water and steam. Each of the regions shows the domain of pressures and temperatures at which a pure phase is in equilibrium. The lines show phase transition temperatures, Tc(P), or phase transition pressures, Pc(T ). The different ways of crossing lines have different names. Ice to water: melting; ice to steam: sublimation; water to steam: boiling; water to ice: freezing; steam to water: condensation; steam to ice: condensation to frost. The transition line from water to steam ends at a point of high pressure and temperature where the two become indistinguishable. At this high pressure steam is compressed till it has a density approaching that of water, and at this high temperature water molecules are energetic like a vapor. This special point is a second-order phase transition point (see Fig. 1.3.8). ❚
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Figure 1.3.8 Similar to Fig. 1.3.6, each of the curves represents the variation of the free energy of a system as a function of macroscopic parameters. In this case, however, the phase transition occurs when two minima emerge from one. This is a second-order phase transition. Below the temperature at which the second-order phase transition occurs, varying the pressure can give rise to a first-order phase transition by changing the relative energies of the two minima (see Figs. 1.3.6 and 1.3.7). ❚
T c +2∆T
T c +∆T
Tc
T c –∆T
T c –2∆T
T c –3∆T
also no first-order transitions. Thus, the second-order transition point occurs as the end of a line of first-order transitions.A second-order transition is found at the end of the liquid-to-vapor phase line of water in Fig. 1.3.7.
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The properties of second-order phase transitions have been extensively studied because of interesting phenomena that are associated with them. Unlike a first-order phase t ransition, there is no coexistence of two phases at the phase transition, because there is only one phase at that point. Instead, there exist large fluctuations in the local properties of the material at the phase transition. A suggestion of why this occurs can be seen from Fig. 1.3.8, where the free energy is seen to be very flat at the phase transition. This results in large excursions (fluctuations) of all the properties of the system except the free energy. These excursions, however, are not coherent over the whole material. Instead, they occur at every length scale from the microscopic on up. The closer a material is to the phase transition, the longer are the length scales that are affected. As the temperature is varied so that the system moves away from the transition temperature,the fluctuations disappear, first on the longest length scales and then on shorter and shorter length scales. Because at the phase transition itself even the macroscopic length scales are affected,thermodynamics itself had to be carefully rethought in the vicinity of second-order phase transitions. The methodology that has b een developed, the renormalization group, is an important tool in the investigation of phase transitions. We will discuss it in Section 1.10. We note that, to be consistent with Question 1.3.1, the specific heat CV must diverge at a second-order phase transition, where energy fluctuations can be large.
1.3.5 Use of thermodynamics and statistical mechanics in describing the real world How do we generalize the notions of thermodynamics that we have just described to apply to more realistic situations? The assumptions of thermodynamics—that systems are in equilibrium and that dividing them into parts leads to unchanged local properties—do not generally apply. The breakdown of the assumptions of thermodynamics occurs for even simple materials, but are more radically violated when we consider biological organisms like trees or people. We still are able to measure their temperature. How do we extend thermodynamics to apply to these systems? We can start by considering a system quite close to the thermodynamic ideal—a pure piece of material that is not in equilibrium. For example, a glass of water in a room. We generally have no trouble placing a thermometer in the glass and measuring the temperature of the water. We know it is not in equilibrium, because if we wait it will evaporate to become a vapor spread out throughout the room (even if we simplify by considering the room closed). Moreover, if we wait longer (a few hundred years to a few tens of thousands of years),the glass itself will flow and cover the table or flow down to the floor, and at least part of it will also sublime to a vapor. The table will undergo its own processes of deterioration. These effects will occur even in an idealized closed room without considerations of various external influences or traffic through the room. There is one essential concept that allows us to continue to apply thermodynamic principles to these materials,and measure the temperature of the water, glass or table, and generally to discover that they are at the same (or close to the same) temperature. The concept is the separation of time scales. This concept is as basic as the other principles of thermodynamics. It plays an essential role in discussions
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of the dynamics of physical systems and in particular of the dynamics of complex systems. The separation of time scales assumes that our observations of systems have a limited time resolution and are performed over a limited time. The processes that occur in a material are then separated into fast processes that are much faster than the time resolution of our observation, slow processes that occur on longer time scales than the duration of observation,and dynamic processes that occur on the time scale of our observation. Macroscopic averages are assumed to be averages over the fast processes. Thermodynamics allows us to deal with the slow and the fast processes but only in very limited ways with the dynamic processes. The dynamic processes are dealt with separately by Newtonian mechanics. Slow processes establish the framework in which thermodynamics can be applied. In formal terms,the ensemble that we use in thermodynamics assumes that all the parameters of the system described by slow processes are fixed. To describe a system using statistical mechanics, we consider all of the slowly varying parameters of the system to be fixed and assume that equilibrium applies to all of the fast processes. Specifically, we assume that all possible arrangements of the fast coordinates exist in the ensemble with a probability given by the Boltzmann probability. Generally, though not always, it is the microscopic processes that are fast. To justify this we can consider that an atom in a solid vibrates at a rate of 1010–1012 times per second,a gas molecule at room temperature travels five hundred meters per second. These are, however, only a couple of select examples. Sometimes we may still choose to perform our analysis by averaging over many possible values of the slow coordinates. When we do this we have two kinds of ensembles—the ensemble of the fast coordinates and the ensemble of the different values of the slow coordinates. These ensembles are called the annealed and quenched ensembles. For example, say we have a glass of water in which there is an ice cube. There are fast processes that correspond to the motion of the water molecules and the vibrations of the ice molecules,and there are also slow processes corresponding to the movement of the ice in the water. Let’s say we want to determine the average amount of ice. If we perform several measurements that determine the coordinates and size of the ice, we may want to average the size we find over all the measurements even though they are measurements corresponding to different locations of the ice. In contrast, if we wanted to measure the motion of the ice, averaging the measurements of location would be absurd. Closely related to the discussion of fast coordinates is the ergodic theorem. The ergodic theorem states that a measurement performed on a system by averaging a property over a long time is the same as taking the average over the ensemble of the fast coordinates. This theorem is used to relate experimental measurements that are assumed to occur over long times to theoretically obtained averages over ensembles. The ergodic theorem is not a theorem in the sense that it has been proven in general, but rather a statement of a property that applies to some macroscopic systems and is known not to apply to others. The objective is to identify when it applies.When it does not apply, the solution is to identify which quantities may be averaged and which may
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not, often by separating fast and slow coordinates or equivalently by identifying quantities conserved by the fast dynamics of the system. Experimental measurements also generally average properties over large regions of space compared to microscopic lengths. It is this spatial averaging rather than time averaging that often enables the ensemble average to stand for experimental measurements when the microscopic processes are not fast compared to the measurement time. For example, materials are often formed of microscopic grains and have many dislocations. The grain boundaries and dislocations do move, but they often change very slowly over time. When experiments are sensitive to their properties, they often average over the effects of many grains and dislocations because they do not have sufficient resolution to see a single grain boundary or dislocation. In order to determine what is the relevant ensemble for a particular experiment, both the effect of time and space averaging must be considered. Technically, this requires an understanding of the correlation in space and time of the properties of an individual system. More conceptually, measurements that are made for particular quantities are in effect made over many independent systems both in space and in time, and therefore correspond to an ensemble average. The existence of correlation is the opposite of independence. The key question (like in the case of the ideal gas) becomes what is the interval of space and time that corresponds to an independent system. These quantities are known as the correlation length and the correlation time. If we are able to describe theoretically the ensemble over a correlation length and correlation time, then by appropriate averaging we can describe the measurement. In summary, the program of use of thermodynamics in the real world is to use the separation of the different time scales to apply equilibrium concepts to the fast degrees of freedom and discuss their influence on the dynamic degrees of freedom while keeping fixed the slow degrees of freedom. The use of ensembles simplifies consideration of these systems by systematizing the use of equilibrium concepts to the fast degrees of freedom.
1.3.6 From thermodynamics to complex systems Our objective in this book is to consider the dynamics of complex systems. While,as discussed in the previous section, we will use the principles of thermodynamics to help us in this analysis,another important reason to review thermodynamics is to recognize what c omplex systems are not. Thermodynamics describes macroscopic systems without structure or dynamics.The task of thermodynamics is to relate the very few macroscopic parameters to each other. It suggests that these are the only relevant parameters in the description of these systems. Materials and complex systems are both formed out of many interacting parts. The ideal gas example described a material where the interaction between the particles was weak.However, thermodynamics also describes solids, where the interaction is strong. Having decided that complex systems are not described fully by the rmodynamics, we must ask, Where do the assumptions of thermodynamics break down? There are several ways the assumptions may break down, and each one is significant and plays a role in our investigation of
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complex systems. Since we have not yet examined particular examples of complex systems, this discussion must be quite abstract. However, it will be useful as we study complex systems to refer back to this discussion. The abstract statements will have concrete realizations when we construct models of complex systems. The assumptions of thermodynamics separate into space-related and timerelated assumptions. The first we discuss is the divisibility of a macroscopic material. Fig. 1.3.2 (page 61) illustrates the property of divisibility. In this process,a small part of a syst em is separated from a large part of the system without affecting the local properties of the material. This is inherent in the use of extensive and intensive quantities. Such divisibility is not true of systems typically considered to be complex systems. Consider, for example, a person as a complex system that cannot be separated and continue to have the same properties. In words, we would say that complex systems are formed out of not only interacting, but also interdependent parts. Since both thermodynamic and complex systems are formed out of interacting parts, it is the concept of interdependency that must distinguish them. We will dedicate a few paragraphs to defining a sense in which “interdependent” can have a more precise meaning. We must first address a simple way in which a system may have a nonextensive energy and still not be a complex system. If we look closely at the properties of a material, say a piece of metal or a cup of water, we discover that its surface is different from the bulk. By separating the material into pieces, the surface area of the material is changed. For macroscopic materials,this generally does not affect the bulk properties of the material.A char acteristic way to identify surface properties, such as the surface energy, is through their dependence on particle number. The surface energy scales as N 2/3, in contrast to the extensive bulk energy that is linear in N. This kind of correction can be incorporated directly in a slightly more detailed treatment of thermodynamics, where every macroscopic parameter has a surface term. The presence of such surface terms is not sufficient to identify a material as a complex system. For this reason, we are careful to identify complex systems by requiring that the scenario of Fig. 1.3.2 is violated by changes in the local (i.e., everywhere including the bulk) properties of the system, rather than just the surface. It may be asked whether the notion of “local properties” is sufficiently well defined as we are using it. In principle,it is not. For now, we adopt this notion from thermodynamics. When only a few properties, like the energy and entropy, are relevant, “affect locally”is a precise concept.Later we would like to replace the use of local thermodynamic properties with a more general concept—the behavior of the system. How is the scenario of Fig. 1.3.2 violated for a complex system? We can find that the local properties of the small part are affected without affecting the local properties of the large part.Or we can find that the local properties of the large part are affected as well. The distinction between these two ways of affecting the system is important, because it can enable us to distinguish between different kinds of complex systems. It will be helpful to name them for later reference. We call the first category of systems complex materials, the second category we call complex organisms.
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Why don’t we also include the possibility that the large part is affected but not the small part? At this point it makes sense to consider generic subdivision rather than special subdivision. By generic subdivision, we mean the ensemble of possible subdivisions rather than a particular one.Once we are considering complex systems,the effect of removal of part of a system may depend on which part is removed. However, when we are trying to understand whether or not we have a complex system, we can limit ourselves to considering the generic effects of removing a part of the system. For this reason we do not consider the possibility that subdivision affects the large system and not the small. This might be possible for the removal of a particular small part, but it would be surprising to discover a system where this is generically true. Two examples may help to illustrate the different classes of complex systems. At least superficially, plants are complex materials, while animals are complex organisms. The reason that plants are complex materials is that the cutting of parts of a plant, such as leaves, a branch, or a root, typically does not affect the local properties of the rest of the plant, but does affect the excised part. For animals this is not generically the case. However, it would be better to argue that plants are in an intermediate category, where some divisions, such as cutting out a lateral section of a tree trunk, affect both small and large parts, while others affect only the smaller part. For animals, essentially all divisions affect both small and large parts.We believe that complex organisms play a special role in the study of complex system behavior. The essential quality of a complex organism is that its properties are tied to the existence of all of its parts. How large is the small part we are talking about? Loss of a few cells from the skin of an animal will not generally affect it. As the size of the removed portion is decreased,it may be expected that the influence on the local properties of the larger system will be reduced. This leads to the concept of a robust complex system. Qualitatively, the larger the part that can be removed from a complex system without affecting its local properties,the more robust the system is. We see that a complex material is the limiting case of a highly robust complex system. The flip side of subdivision of a system is aggregation. For thermodynamic systems, subdivision and aggregation are the same, but for complex systems they are quite different. One of the questions that will concern us is what happens when we place a few or many complex systems together. Generally we expect that the individual complex systems will interact with each other. However, one of the points we can make at this time is that just placing together many complex systems, trees or people, does not make a larger complex system by the criteria of subdivision. Thus, a collection of complex systems may result in a system that behaves as a thermodynamic system under subdivision—separating it into parts does not affect the behavior of the parts. The topic of bringing together many pieces or subdividing into many parts is also quite distinct from the topic of subdivision by removal of a single part. This brings us to a second assumption we will discuss.Thermodynamic systems are assumed to be composed of a very large number of particles. What about complex systems? We know that the number of molecules in a cup of water is not greater than the number of molecules
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in a human being.And yet,we understand that this is not quite the right point.We should not be counting the number of water molecules in the person,instead we might count the number of cells, which is much smaller. Thus appears the problem of counting the number of components of a system. In the context of correlations in materials, this was briefly discussed at the end of the last section. Let us assume for the moment that we know how to count the number of components. It seems clear that systems with only a few components should not be treated by thermodynamics.One of the interesting questions we will discuss is whether in the limit of a very large number of components we will always have a thermodynamic system.Stated in a simpler way from the point of view of the study of complex systems, the question becomes how large is too large or how many is too many. From the thermodynamic perspective the question is, Under what circumstances do we end up with the thermodynamic limit? We now switch to a discussion of time-related assumptions.One of the basic assumptions of thermodynamics is the ergodic theorem that enables the description of a single system using an ensemble. When the ergodic theorem breaks down, as discussed in the previous section, additional fixed or quenched variables become important. This is the same as saying that there are significant differences between different examples of the macroscopic system we are interested in. This is a necessary condition for the existence of a complex system. The alternative would be that all realizations of the system would be the same, which does not coincide with intuitive notions of complexity. We will discuss several examples of the breaking of the ergodic theorem later. The simplest example is a magnet. The orientation of the magnet is an additional parameter that must be specified, and therefore the ergodic theorem is violated for this system. Any system that breaks symmetry violates the ergodic theorem. However, we do not accept a magnet as a complex system. Therefore we can assume that the breaking of ergodicity is a necessary but not sufficient condition for complexity. All of the systems we will discuss break ergodicity, and therefore it is always necessary to specify which coordinates of the complex system are fixed and which are to be assumed to be so rapidly varying that they can be assigned equilibrium Boltzmann probabilities. A special case of the breaking of the ergodic theorem, but one that strikes even more deeply at the assump tions of thermodynamics, is a violation of the separation of time scales. If there are dynamical processes that occur on every time scale, then it becomes impossible to treat the system using the conventional separation of scales into fast,slow and dynamic processes.As we will discuss in Section 1.10,the techniques of renormalization that are used in phase transitions to deal with the existence of many spatial scales may also be used to describe systems changing on many time scales. Finally, inherent in thermodynamics,the concept of equilibrium and the ergodic theorem is the assumption that the initial condition of the system does not matter. For a complex system,the initial condition of the system does matter over the time scales relevant to our observation. This brings us back to the concept of correlation time. The correlation time describes the length of time over which the initial conditions are relevant to the dynamics. This means that our observation of a complex system must be shorter than a correlation time.The spatial analog, the correlation length,describes
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the effects of surfaces on the system. The discussion of the effects of subdivision also implies that the system must be smaller than a correlation length. This means that complex systems change their internal structure—adapt—to conditions at their boundaries. Thus, a suggestive though incomplete summary of our discussion of complexity in the context of thermodynamics is that a complex system is contained within a single correlation distance and correlation time.
1.4
Activated Processes (and Glasses)
In the last section we saw figures (Fig. 1.3.7) showing the free energy as a function of a macroscopic parameter with two minima. In this section we analyze a single particle system that has a potential energy with a similar shape (Fig. 1.4.1). The particle is in equilibrium with a thermal reservoir. If the average energy is lower than the energy of the barrier between the two wells, then the particle generally resides for a time in one well and then switches to the other. At very low temperatures, in equilibrium,it will be more and more likely to be in the lower well and less likely to be in the higher well. We use this model to think about a system with two possible states, where one state is higher in energy than the other. If we start the system in the higher energy state, the system will relax to the lower energy state. Because the process of relaxation is enabled or accelerated by energy from the thermal resevoir, we say that it is activated.
E
EB
EB
E1
E1
E –1 x –1
xB
x1
x
Figure 1.4.1 Illustration of the potential energy of a system that has two local minimum energy configurations x1 and x−1. When the temperature is lower than the energy barriers EB − E−1 and EB − E1, the system may be considered as a two-state system with transitions between them. The relative probability of the two states varies with temperature and the relative energy of the bottom of the two wells. The rate of transition also varies with temperature. When the system is cooled systematically the two-state system is a simple model of a glass (Fig. 1.4.2). At low temperatures the system can not move from one well to the other, but is in equilibrium within a single well. ❚
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1.4.1 Two-state systems It might seem that a system with only two different states would be easy to analyze. Eventually we will reach a simple problem. However, building the simple model will require us to identify some questions and approximations relevant to our understanding of the application of this model to physical systems (e.g. the problem of protein folding found in Chapter 4). Rather than jumping to the simple two-state problem (Eq. (1.4.40) below), we begin from a particle in a double-well potential. The kinetics and thermodynamics in this system give some additional content to the thermodynamic discussion of the previous section and introduce new concepts. We consider Fig. 1.4.1 as describing the potential energy V(x) experienced by a classical particle in one dimension. The region to the right of xB is called the right well and to the left is called the left well.A classical trajectory of the particle with conserved energy would consist of the particle bouncing back and forth within the potential well between two points that are the solution of the equation V(x) = E, where E is the total energy of the particle. The kinetic energy at any time is given by E(x, p)−V (x) = 21 mv 2
(1.4.1)
which determines the magnitude of the velocity at any position but not the direction. The velocity switches direction every bounce.When the energy is larger than EB , there is only one distinct trajectory at each energy. For energies larger than E1 but smaller than EB , there are two possible trajectories, one in the right well—to the right of xB — and one in the left well. Below E1, which is the minimum energy of the right well,there is again only one trajectory possible, in the left well. Below E−1 there are no possible locations for the particle. If we consider this system in isolation,there is no possibility that the particle will change from one trajectory to another. Our first objective is to enable the particle to be in contact with some other system (or coordinate) with which it can transfer energy and momentum. For example, we could imagine that the particle is one of many moving in the double well—like the ideal gas. Sometimes there are collisions that change the energy and direction of the motion. The same effect would be found for many other ways we could imagine the particle interacting with other systems. The main approximation, however, is that the interaction of the particle with the rest of the universe occurs only over short times. Most of the time it acts as if it were by itself in the potential well. The particle follows a trajectory and has an energy that is the sum of its kinetic and potential energies (Eq.(1.4.1)). There is no need to describe the energy associated with the interaction with the other systems. All of the other particles of the gas (or whatever picture we imagine) form the thermal reservoir, which has a well-defined temperature T. We can increase the rate of collisions between the system and the reservoir without changing our description. Then the particle does not go very far before it forgets the direction it was traveling in and the energy that it had. But as long as the collisions themselves occur over a short time compared to the time between collisions,any time we look at the particle, it has a well-defined energy and momentum. From moment
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to moment,the kinetic energy and momentum changes unpredictably. Still,the position of the particle must change continuously in time. This scenario is known as diffusive motion. The different times are related by: collision (interaction) time << time between collisions << transit time where the transit time is the time between bounces from the walls of the potential well if there were no collisions—the period of oscillation of a particle in the well. The particle undergoes a kind of random walk, with its direction and velocity changing randomly from moment to moment. We will assume this scenario in our treatment of this system. When the par ticle is in contact with a thermal reservoir, the laws of thermodynamics apply. The Boltzmann probability gives the probability that the particle is found at position x with momentum p:
Z=
∑ x ,p
P(x, p) = e −E(x ,p)/kT / Z 1 e −E(x ,p )/kT = dxdp e −E(x ,p )/kT h
∫
(1.4.2)
Formally, this expression describes a large number of independent systems that make up a canonical ensemble.The ensemble of systems provides a formally precise way of describing probabilities as the number of systems in the ensemble with a particular value of the position and momentum. As in the previous section, Z guarantees that the sum over all probabilities is 1. The factor of h is not relevant in what follows, but for completeness we keep it and associate it with the momentum integral, so that Σp → ∫dp/h. If we are interested in the position of the particle,and are not interested in its momentum, we can simplify this expression by integrating over all values of the momentum. Since the energy separates into kinetic and potential energy: P(x) =
∫
e −V(x )/kT (dp /h) e − p
2
/ 2mkT
∫dx e −V(x )/kT ∫ (dp /h) e − p
2
/2mkT
=
e −V(x)/kT
∫ dx e
−V (x )/kT
(1.4.3)
The resulting expression looks similar to our original expression. Its meaning is somewhat different,however, because V(x) is only the potential energy of the system. Since the kinetic energy contributes equivalently to the probability at every location, V(x) determines the probability at every x. An expression of the form e−E/kT is known as the Boltzmann factor of E. Thus Eq.(1.4.3) says that the probability P(x) is proportional to the Boltzmann factor of V(x). We will use this same trick to describe the probability of being to the right or being to the left of xB in terms of the minimum energy of each well. To simplify to a two-state system, we must define a variable that specifies only which of the two wells the particle is in. So we label the system by s = ±1, where s = +1 if x > xB and s = −1 if x < xB for a par ticular realization of the system at a particular time, or:
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s = sign(x − xB)
(1.4.4)
Probabilistically, the case x = xB never happens and therefore does not have to be accounted for. We can calculate the probability P(s) of the system having a value of s =+1 using: ∞
P(1) =
∫ dx e
−V (x )/kT
∫ dx e
−V (x)/ kT
(1.4.5)
xB
The largest contribution to this probability occurs when V(x) is smallest. We assume that kT is small compared to EB , then the value of the integral is dominated by the region immediately in the vicinity of the minimum energy. Describing this as a two-state system is only meaningful when this is true. We simplify the integral by expanding it in the vicinity of the minimum energy and keeping only the quadratic term: V (x) = E 1 + 12 m
2 2 1 (x − x1 )
+ … =E1 + 21 k1 (x − x 1 )2 + …
(1.4.6)
where k1 = m
2 1
=
d 2V(x) dx 2
(1.4.7) x1
is the effective spring constant and 1 is the frequency of small oscillations in the right well. We can now write Eq. (1.4.5) in the form ∞
∫
e −E1 /kT dx e −k1 (x −x 1 ) P(1) =
2
/2 kT
(1.4.8)
xB
∫ dx e
−V (x )/kT
Because the integrand in the numerator falls rapidly away from the point x = x1, we could extend the lower limit to −∞. Similarly, the probability of being in the left well is: xB
∫
e −E −1 /kT dx e −k−1 (x −x −1 ) P(−1) =
2
/2kT
(1.4.9)
−∞
∫
dx e −V (x)/kT
Here the upper limit of the integral could be extended to ∞. It is simplest to assume that k1 = k−1. This assumption, that the shape of the wells are the same, does not significantly affect most of the discussion (Question 1.4.1–1.4.2). The two probabilities are proportional to a new constant times the Boltzmann factor e−E/kT of the energy at the bottom of the well. This can be seen e ven without performing the integrals in Eq. (1.4.8) and Eq. (1.4.9). We redefine Z for the two-state representation:
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P(−1) =
P(1) =
99
e −E −1 /kT Zs
(1.4.10)
e −E1 /kT Zs
(1.4.11)
The new normalization Zs can be obtained from: P(1)+ P(−1) = 1
(1.4.12)
Z s = e −E1 /kT + e −E −1 / kT
(1.4.13)
giving which is different from the value in Eq. (1.4.2). We arrive at the desired two-state result: P(1) =
e −E1 /kT e
−E1 /kT
+e
1
=
−E−1 /kT
1+ e
(E1 −E−1 )/kT
= f (E 1 − E −1 )
(1.4.14)
where f is the Fermi probability or Fermi function: f (x) =
1
(1.4.15)
1 +e x /kT
For readers who were introduced to the Fermi function in quantum statistics,it is not unique to that field, it occurs anytime there are exactly two different possibilities. Similarly, P(−1) =
e −E−1 /kT e
−E 1 / kT
+e
=
−E −1 /kT
1 1 +e
(E −1 −E1 )/ kT
= f (E −1 − E1 )
(1.4.16)
which is consistent with Eq. (1.4.12) above since f (x) + f (−x) = 1
(1.4.17)
Q
uestion 1.4.1 Discuss how k 1 ≠ k−1 would affect the results for the twostate system in equilibrium. Obtain expressions for the probabilities in each of the wells. Solution 1.4.1 Extending the integrals to ±∞, as described in the text after Eq. (1.4.8) and Eq. (1.4.9), we obtain: P(1) =
P(−1) =
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e −E1 /kT 2 kT /k 1
∫
dx e −V (x)/ kT
e −E1 / kT 2 kT /k −1
∫ dx e
−V (x )/ kT
(1.4.18)
(1.4.19)
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Because of the approximate extension of the integrals, we are no longer guaranteed that the sum of these probabilities is 1. However, within the accuracy of the approximation, we can reimpose the normalization condition. Before we do so, we choose to rewrite k1 = m 12 = m(2 1)2, where 1 is the natural frequency of the well. We then ignore all common factors in the two probabilities and write P(1) =
P(−1) =
−1 −E 1 /kT 1 e
−1 −E−1 /kT −1 e
(1.4.21)
Z s′
−1 −E1 /kT −1e
Z s′ =
(1.4.20)
Z s′
+
−1 −E−1 /kT −1 e
(1.4.22)
Or we can write, as in Eq. (1.4.14) P(1) =
1 1+ ( 1 /
−1 )e
(E1 −E−1 )/kT
(1.4.23)
and similarly for P(−1). ❚
Q
uestion 1.4.2 Redefine the energies E1 and E−1 to include the effect of the difference between k1 and k−1 so that the probability P(1) (Eq. (1.4.23)) can be written like Eq. (1.4.14) with the new energies. How is the result related to the concept of free energy and entropy? Solution 1.4.2 We define the new energy of the right well as F1 = E 1 +kT ln( 1 )
(1.4.24)
This definition can be seen to recover Eq. (1.4.23) from the form of Eq. (1.4.14) as P(1) = f (F1 − F−1 )
(1.4.25)
Eq. (1.4.24) is very reminiscent of the definition of the free energy Eq. (1.3.33) if we use the expression for the entropy: S1 = −k ln( 1 )
(1.4.26)
Note that if we consider the temperature dependence, Eq. (1.4.25) is not identical in its behavior with Eq.(1.4.14). The free energy, F1, depends on T, while the energy at the bottom of the well, E1, does not. ❚ In Question 1.4.2, Eq. (1.4.24), we have defined what might be interpreted as a free energy of the right well. In Section 1.3 we defined only the free energy of the system as a whole. The new free energy is for part of the ensemble rather than the whole ensemble. We can do this quite generally. Start by identifying a certain subset of all
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possible states of a system. For example, s = 1 in Eq. (1.4.4). Then we define the free energy using the expression: Fs (1) = −kT ln(
∑
s ,1
e −E({x ,p })/ kT ) = −kT ln(Z 1)
{x,p}
(1.4.27)
This is similar to the usual expression for the free energy in terms of the partition function Z, but the sum is only over the subset of states. When there is no ambiguity, we often drop the subscript and write this asF(1). From this definition we see that the probability of being in the subset of states is proportional to the Boltzmann factor of the free energy P(1) ∝e −Fs (1)/ kT
(1.4.28)
If we have several different subsets that account for all possibilities, then we can normalize Eq. (1.4.28) to find the probability itself. If we do this for the left and right wells, we immediately arrive at the expression for the probabilities in Eq.(1.4.14) and Eq. (1.4.16), with E1 and E−1 replaced by Fs(1) and Fs(−1) respectively. From Eq.(1.4.28) we see that for a collection of states,the free energy plays the same role as the energy in the Boltzmann probability. We note that Eq. (1.4.24) is not the same as Eq.(1.4.27). However, as long as the relative energy is the same, F1 − F−1 = Fs(1) − Fs(−1),the normalized probability is unchanged. When k1 = k−1, the entropic part of the free energy is the same for both wells. Then direct use of the energy instead of the free energy is valid,as in Eq.(1.4.14). We can evaluate the free energy of Eq. (1.4.27), including the momentum integral: ∞
∞
∫ ∫
∫
∫
Z 1 = dx (dp /h) e −E(x ,p)/ kT = dxe −V (x)/kT (dp /h) e −p xB
xB
∞
∫
≈ e −E1 /kT dx e −k1 (x −x 1 )
2
/2 kT
2
/ 2mkT
2 mkT /h ≈ e −E1 / kT m /k 1 2 kT /h
(1.4.29)
xB
= e −E1 /kT kT /h
1
Fs (1) = E1 +kT ln(h
1 /kT )
(1.4.30)
where we have used the definition of the well oscillation frequency above Eq.(1.4.20) to simplify the expression.A similar expression holds for Z−1. The result would be exact for a pure harmonic well. The new definition of the free energy of a set of states can also be used to understand the treatment of macroscopic systems,specifically to explain why the energy is determined by minimizing the free energy. Partition the possible microstates by the value of the energy, as in Eq. (1.3.35). Define the free energy as a function of the energy analogous to Eq. (1.4.27) −E( {x ,p }) / kT (1.4.31) F(U) = −kT ln E( {x ,p} ),U e {x ,p }
∑
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Since the relative probability of each value of the energy is given by P(U) ∝e −F(U )/kT
(1.4.32)
the most likely energy is given by the lowest free energy. For a macroscopic system, the most likely value is so much more likely than any other value that it is observed in any measurement. This can immediately be generalized. The minimization of the free energy gives not only the value of the energy but the value of any macroscopic parameter.
1.4.2 Relaxation of a two-state system To investigate the kinetics of the two-state system, we assume an ensemble of systems that is not an equilibrium ensemble. Instead,the ensemble is characterized by a timedependent probability of occupying the two wells: P(1) → P(1;t ) P(−1) → P(−1;t )
(1.4.33)
Normalization continues to hold at every time: P(1;t) + P(−1;t ) = 1
(1.4.34)
For example, we might consider starting a system in the up per well and see how the system evolves in time. Or we might consider starting a system in the lower well and see how the system evolves in time. We answer the question using the time-evolving probabilities that describe an ensemble of systems with the same starting condition. To achieve this objective, we construct a differential equation describing the rate of change of the probability of being in a particular well in terms of the rate at which systems move from one well to the other. This is just the Master equation approach from Section 1.2.4. The systems that make transitions from the left to the right well are the ones that cross the point x = xB. More precisely, the rate at which transitions occur is the probability current per unit time of systems at xB , moving toward the right. Similar to Eq. (1.3.47) used to obtain the pressure of an ideal gas on a wall,the number of particles crossing xB is the probability of systems at xB with velocity v, times their velocity: ∞
∫
J(1 |−1) = (dp /h)vP(x B , p;t)
(1.4.35)
0
where J(1|−1) is the number of systems per unit time moving from the left to the right. There is a hidden assumption in Eq. (1.4.35). We have adopted a notation that treats all systems on the left together. When we are considering transitions,this is only valid if a system that crosses x = xB from right to left makes it down into the well on the left, and thus does not immediately cross back over to the side it came from. We further assume that in each well the systems are in equilibrium, even when the two wells are not in equilibrium with each other. This means that the probability of being in a particular location in the right well is given by:
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103
P(x, p;t) = P(1;t)e −E (x ,p )/kT /Z1 ∞
∫
Z 1 = dxdp e −E(x,p)/ kT
(1.4.36)
xB
In equilibrium,this statement is true because then P(1) = Z1 /Z. Eq.(1.4.36) presumes that the rate of collisions between the particle and the thermal reservoir is faster than both the rate at which the system goes from one well to the other and the frequency of oscillation in a well. In order to evaluate the transition rate Eq.(1.4.35), we need the probability at xB. We assume that the systems that cross xB moving from the left well to the right well (i.e.,moving to the right) are in equilibrium with systems in the left well from where they came. Systems that are moving from the right well to the left have the e quilibrium distribution characteristic of the right well. With these assumptions, the rate at which systems hop from the left to the right is given by: ∞
2 J(1 |−1) = (dp /h)(p /m)P(−1;t )e −(EB +p / 2m)/ kT /Z −1
∫
0
= P(−1;t )e −E B / kT (kT /h)/Z −1
(1.4.37)
We find using Eq. (1.4.29) that the current of systems can be written in terms of a transition rate per system: J(1| −1) = R(1| −1)P(−1;t ) R(1| −1) = e −(EB −E −1 ) /kT
(1.4.38)
−1
Similarly, the current and rate at which systems hop from the right to the left are given by: J(−1 |1) = R(−1 |1)P(1;t ) R(−1 |1) = 1e −(E B −E1 )/ kT
(1.4.39)
When k1 = k−1 then 1 = −1. We continue to deal with this case for simplicity and define = 1 = −1. The expressions for the rate of transition suggest the interpretation that the frequency is the rate of attempt to cross the barrier. The probability of crossing in each attempt is given by the Boltzmann factor, which gives the likelihood that the energy exceeds the barrier. While this interpretation is appealing, and is often given,it is misleading. It is better to consider the frequency as describing the width of the well in which the particle wanders. The wider the well is,the less likely is a barrier crossing. This interpretation survives better when more general cases are considered. The transition rates enable us to construct the time variation of the probability of occupying each of the wells. This gives us the coupled equations for the two probabilities: P˙ (1;t) = R(1| −1)P(−1;t ) − R(−1|1)P(1;t ) (1.4.40) P˙ (−1;t ) = R(−1|1)P(1;t ) − R(1| −1)P(−1;t )
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These are the Master equations (Eq. (1.2.86)) for the two-state system. We have arrived at these equations by introducing a set of assumptions for treating the kinetics of a single particle. The equations are much more general, since they say only that there is a rate of transition between one state of the system and the other. It is the correspondence between the two-state system and the moving particle that we have established in Eqs. (1.4.38) and (1.4.39). This correspondence is approximate. Eq. (1.4.40) does not rely upon the relationship between EB and the rate at which systems move from one well to the other. However, it does rely upon the assumption that we need to know only which well the system is in to specify its rate of transition to the other well. On average this is always true, but it would not be a good description of the system, for example, if energy is conserved and the key question determining the kinetics is whether the particle has more or less energy than the barrier EB. We can solve the coupled equations in Eq. (1.4.40) directly. Both equations are not necessary, given the normalization constraint Eq.(1.4.34). Substituting P(−1;t) = 1 − P(1;t) we have the equation P˙ (1;t) = R(−1 |1) − P(1;t )(R(1 |−1)+ R(−1|1))
(1.4.41)
We can rewrite this in terms of the equilibrium value of the probability. By definition this is the value at which the time derivative vanishes. P(1;∞) = R(−1| 1) /(R(1| −1) + R(−1|1)) = f (E 1 − E −1)
(1.4.42)
where the right-hand side follows from Eq.(1.4.38) and Eq.(1.4.39) and is consistent with Eq. (1.4.13), as it must be. Using this expression, Eq. (1.4.24) becomes P˙ (1;t) =(P(1;∞)− P(1;t ))/ (1.4.43) where we have defined an additional quantity 1/ = (R(1| −1) + R(−1 |1)) = (e −(E B −E1 )/ kT +e −(EB −E−1 )/kT )
(1.4.44)
The solution of Eq. (1.4.43) is P(1;t) =(P(1;0)− P(1;∞))e −t / + P(1;∞)
(1.4.45)
This solution describes a decaying exponential that changes the probability from the starting value to the equilibrium value. This explains the definition of , called the relaxation time. Since it is inversely related to the sum of the rates of transition between the wells,it is a typical time taken by a system to hop between the wells. The relaxation time does not depend on the starting probability. We note that the solution of Eq.(1.4.41) does not depend on the explicit form of P(1; ∞) or . The definitions implied by the first equal signs in Eq.(1.4.42) and Eq.(1.4.44) are sufficient. Also, as can be quickly checked, we can replace the index 1 with the index −1 without changing anything else in Eq (1.4.45). The other equations are valid (by symmetry) after the substitution 1 ↔ −1.
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105
There are several intuitive relationships between the equilibrium probabilities and the transition rates that may be written down. The first is that the ratio of the equilibrium probabilities is the ratio of the transition rates: P1(∞) P−1 (∞) = R(−1|1)/R(1| −1)
(1.4.46)
The second is that the equilibrium probability divided by the relaxation time is the rate of transition: P1(∞)
= R(−1|1)
(1.4.47)
Q
uestion 1.4.3 Eq. (1.4.45) implies that the relaxation time of the system depends largely on the smaller of the two energy barriers EB − E1 and EB − E−1. For Fig. 1.4.1 the smaller barrier is EB − E1. Since the relaxation time is independent of the starting probability, this barrier controls the rate of relaxation whether we start the system from the lower well or the upper well. Why does the barrier EB − E1 control the relaxation rate when we start from the lower well? Solution 1.4.3 Even though the rate of transition from the lower well to the upper well is controlled by EB − E−1, the fraction of the ensemble that must make the transition in order to reach equilibrium depends on E1. The higher it is,the fewer systems must make the transition from s = −1 to s = 1. Taking this into consideration implies that the time to reach equilibrium depends on EB − E1 rather than EB − E−1. ❚
1.4.3 Glass transition Glasses are materials that when cooled from the liquid do not undergo a conventional transition to a solid. Instead their viscosity increases,and in the vicinity of a particular temperature it becomes so large that on a reasonable time scale they can be treated as solids. However, on long enough time scales,they flow as liquids. We will model the glass transition using a two-state system by considering what happens as we cool down the two-state system. At high enough temperatures, the system hops back and forth between the two minima with rates given by Eqs.(1.4.38) and (1.4.39). is a microscopic quantity; it might be a vibration rate in the material. Even if the barriers are higher than the temperature, EB − E±1 >> kT, the system will still be able to hop back and forth quite rapidly from a macroscopic perspective. As the system is cooled down, the hopping back and forth slows down. At some point the rate of hopping will become longer than the time we are observing the system. Systems in the higher well will stay there. Systems in the lower well will stay there. This means that the population in each well becomes fixed. Even when we continue to cool the system down, there will be no change, and the ensemble will no longer be in equilibrium. Within each well the system will continue to have a probability distribution for its energy given by the Boltzmann probability, but the relative
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populations of the two wells will no longer be described by the equilibrium Boltzmann probability. To gain a feeling for the numbers,a typical atomic vibration rate is 1012 /sec. For a barrier of 1eV, at twice room temperature, kT ≈ 0.05eV (600°K), the transition rate would be of order 10 3/sec. This is quite slow from a microscopic perspective, but at room temperature it would be only 10−6/sec, or one transition per year. The rate at which we cool the system down plays an essential role. If we cool faster, then the temperature at which transitions stop is higher. If we cool at a slower rate, then the temperature at which the transitions stop is lower. This is found to be the case f or glass transitions, where the cooling rate determines the departure point from the equilibrium trajectory of the system,and the eventual properties of the glass are also determined by the cooling rate. Rapid cooling is called quenching. If we raise the temperature and lower it slowly, the procedure is called annealing. Using the model two-state system we can simulate what would happen if we perform an experiment of cooling a system that becomes a glass.Fig. 1.4.2 shows the probability of being in the upper well as a function of the temperature as the system is cooled down. The curves depart from the equilibrium curve in the vicinity of a transition temperature we might call a freezing transition, because the kinetics become frozen. The glass transition is not a transition like a first- or second-order transition (Section 1.3.4) because it is a transition of the kinetics rather than of the equilibrium structure of the system. Below the freezing transition, the relative probability of the system being in the upper well is given approximately by the equilibrium probability at the transition. The freezing transition of the relative population of the upper state and the lower state is only a simple model of the glass transition;however, it is also more widely applicable. The freezing does not depend on cooperative effects of many particles. To find examples, a natural place to look is the dynamics of individual atoms in solids. Potential energies with two wells occur for impurities, defects and even bulk atoms in a solid. Impurities may have two different local configurations that differ in energ y and are separated by a barrier. This is a direct analog of our model two-state system. When the temperature is lowered, the relative population of the two configurations becomes frozen. If we raise the temperature, the system can equilibrate again. It is also possible to artificially cause impurity configurations to have unequal energies.One way is to apply uniaxial stress to a crystal—squeezing it along one axis. If an impurity resides in a bond between two bulk atoms, applying stress will raise the energy of impurities in bonds oriented with the stress axis compared to bonds perpendicular to the stress axis. If we start at a relatively high temperature, apply stress and then cool down the material, we can freeze unequal populations of the impurity. If we have a way of measuring relaxation, then by raising the temperature gradually and observing when the defects begin to equilibrate we can discover the barrier to relaxation. This is one of the few methods available to study the kinetics of impurity reorientation in solids. The two-state system provides us with an example of how a simple system may not be able to equilibrate over experimental time scales. It also shows how an e qui-
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0.12 P(1;t) t
0.10 0.08 0.06 0.04
200˚K/sec 100˚K/sec
…
0.4˚K/sec
0.02 P(1;∞) 0.00 100
200
300
400
500
T
600
Figure 1.4.2 Plot of the fraction of the systems in the higher energy well as a function of temperature. The equilibrium value is shown with the dashed line. The solid lines show what happens when the system is cooled from a high temperature at a particular cooling rate. The example given uses E1 − E−1 = 0.1eV and EB − E−1 = 1.0eV. Both wells have oscillation frequencies of v = 1012/sec. The fastest cooling rate is 200°K/sec and each successive curve is cooled at a rate that is half as fast, with the slowest rate being 0.4°K/sec. For every cooling rate the system stops making transitions between the wells at a particular temperature that is analogous to a glass transition in this system. Below this temperature the probability becomes essentially fixed. ❚
librium ensemble can be used to treat relative probabilities within a subset of states. Because the motion within a particular well is fast,the relative probabilities of different positions or momenta within a well may be described using the Boltzmann probability. At the same time, the relative probability of finding a system in each of the two wells depends on the initial conditions and the history of the system—what temperature the system experienced and for how long. At sufficiently low temperatures, this relative probability may be treated as fixed. Systems that are in the higher well may be assumed to stay there. At intermediate temperatures, a treatment of the dynamics of the transition between the two wells can (and must) be included. This manifests a violation of the ergodic theorem due to the divergence of the time scale
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Introduction and Preliminaries
for equilibration between the two wells. Thus we have identified many of the features that are necessary in describing nonequilibrium systems: divergent time scales, violation of the ergodic theorem, frozen and dynamic coordinates. We have illustrated a method for treating systems where there is a separation of long time scales and short time scales.
Q
uestion 1.4.4 Write a program that can generate the time dependence of the two-state system for a specified time history. Reproduce Fig. 1.4.2. For an additional “experiment,” try the following quenching and annealing sequence: a. Starting from a high enough temperature to be in equilibrium, cool the system at a rate of 10°K/sec down to T = 0. b. Heat the system up to temperature Ta and keep it there for one second. c. Cool the system back down to T = 0 at rate of 100°K/sec. Plot the results as a function of Ta. Describe and explain them in words. ❚
1.4.4 Diffusion In this section we briefly consider a multiwell system. An example is illustrated in Fig. 1.4.3, where the potential well depths and barriers vary from site to site. A simpler case is found in Fig. 1.4.4, where all the well depths and barriers are the same. A concrete example would be an interstitial impurity in an ideal crystal. The impurity lives in a periodic energy that repeats every integral multiple of an elementary length a. We can apply the same analysis from the previous section to describe what happens to a system that begins from a particular well at x = 0. Over time, the system makes transitions left and right at random,in a manner that is reminiscent of a random walk. We will see in a moment that the connection with the random walk is valid but requires some additional discussion.
V(x) E B (0|-1)
E –1
E B (2|1)
E2
E1
E0
-1
E B (3|2)
E B (1|0)
0
1
2
x
Figure 1.4.3 Illustration of a multiple-well system with barrier heights and well depths that vary from site to site. We focus on the uniform system in Fig. 1.4.4. ❚
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The probability of the system being in a particular well is changed by probability currents into the well and out from the well. Systems can move to or from the well immediately to their right and immediately to their left. The Master equation for the ith well in Fig. 1.4.3 is: P˙ (i;t ) = R(i |i − 1)P(i − 1;t) + R(i |i +1)P(i +1;t ) −(R(i + 1|i) + R(i −1 |i))P(i;t ) (1.4.48) R(i + 1|i) = ie −(E B (i +1|i )−Ei )/kT
(1.4.49)
R(i − 1|i) = ie −(EB (i|i−1)−Ei )/kT
where Ei is the depth of the ith well and EB(i + 1|i) is the barrier to its right. For the periodic system of Fig. 1.4.4 ( i → , EB(i + 1|i) → EB) this simplifies to: P˙ (i;t ) = R(P(i −1;t )+ P(i + 1;t ) − 2P(i ;t )) (1.4.50) R = e − (E
B − E 0) /kT
(1.4.51)
Since we are already describing a continuum differential equation in time,it is convenient to consider long times and write a continuum equation in space as well. Allowing a change in notation we write P(i;t ) → P(x i ;t )
(1.4.52)
Introducing the elementary distance between wells a we can rewrite Eq. (1.4.50) using: (P(i −1;t )+ P(i + 1;t ) − 2P(i;t )) →
a2 (P(x i − a;t )+ P(x i + a;t) − 2P(x i ;t )) a2
(1.4.53)
2
→
x
P(x ;t ) 2
V(x) EB
E0
-1
0
1
2
x
Figure 1.4.4 When the barrier heights and well depths are the same, as illustrated, the long time behavior of this system is described by the diffusion equation. The evolution of the system is controlled by hopping events from one well to the other. The net effect over long times is the same as for the random walk discussed in Section 1.2. ❚
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Introduction and Preliminaries
where the last expression assumes a is small on the scale of interest. Thus the continuum version of Eq. (1.4.50) is the conventional diffusion equation: 2
P˙ (x;t) = D
x2
(1.4.54)
P(x;t)
The diffusion constant D is given by: D = a 2 R = a 2 e −(EB −E0 )/ kT
(1.4.55)
The solution of the diffusion equation, Eq. (1.4.54), depends on the initial conditions that are chosen. If we consider an ensemble of a system that starts in one well and spreads out over time, the solution can be checked by substitution to be the Gaussian distribution found for the random walk in Section 1.2: P(x,t) =
1 4 Dt
e −x
2
/4 Dt
=
1 2
e −x
2
/2
2
(1.4.56)
= 2Dt We see that motion in a set of uniform wells after a long time reduces to that of a random walk. How does the similarity to the random walk arise? This might appear to be a natural result,since we showed that the Gaussian distribution is quite general using the central limit theorem. The scenario here ,h owever, is quite different. The central limit theorem was proven in Section 1.2.2 for the case of a distribution of probabilities of steps taken at specific time intervals. Here we have a time continuum. Hopping events may happen at any time. Consider the case where we start from a particular well. Our differential equation describes a system that might hop to the next well at any time. A hop is an event, and we might concern ourselves with the distribution of such events in time. We have assumed that these events are uncorrelated.There are unphysical consequences of this assumption. For example, no matter how small an interval of time we choose,the particle has some probability of traveling arbitrarily far away. This is not necessarily a correct microscopic picture, but it is the continuum model we have developed. There is a procedure to convert the event-controlled hopping motion between wells into a random walk that takes steps with a certain probability at specific time intervals. We must select a time interval. For this time interval, we evaluate the total probability that hops move a system from its original position to all possible positions of the system. This would give us the function f (s) in Eq.(1.2.34). As long as the mean square displacement is finite,the central limit theorem continues to apply to the probability distribution after a long enough time. The generality of the conclusion also implies that the result is more widely applicable than the assumptions indicate.However, there is a counter example in Question 1.4.5.
Q
uestion 1.4.5 Discuss the case of a particle that is not in contact with a thermal resevoir moving in the multiple well system (energy is conserved).
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Solution 1.4.5 If the energy of the system is lower than EB , the system stays in a single well bouncing back and forth. A model that describes how t ransitions occur between wells would just say there are none. For the case where the energy is larger than EB, the system will move with a periodically varying velocity in one direction. There is a problem in selecting an ensemble to describe it. If we choose the ensemble with only one system moving in one direction, then it is described as a deterministic walk. This description is consistent with the motion of the system. However, we might also think to describe the system using an ensemble consisting of particles with the same energy. In this case it would be one particle moving to the right and one moving to the left. Taking an interval of time to be the time needed to move to the next well, we would find a transition probability of 1/2 to move to the right and the same to the left. This would lead to a conventional random walk and will give us an incorrect result for all later times. This example illustrates the need for an assumption that has not yet been explicitly mentioned. The ensemble must describe systems that can make transitions to each other. Since the energy-conserving systems cannot switch directions, the ensemble cannot include both directions. It is enough, however, for there to be a small nonzero probability for the system to switch directions for the central limit theorem to apply. This means that over long enough times, the distribution will be Gaussian. Over short times,however, the probability distribution from the random walk model and an almost ballistic system would not be very similar. ❚ We can generalize the multiple well pic ture to describe a biased random walk. The potential we would use is a “washboard potential,” illustrated in Fig. 1.4.5. The Master equation is: P˙ (i;t ) = R+ P(i − 1;t) + R −P(i +1;t )− (R + + R− )P(i;t ) R + = ie −
R − = ie −
(1.4.57)
E+ /kT
(1.4.58)
E− /kT
To obtain the continuum limit, replace i → x : P(i + 1;t) → P(x + a,t), and P(i − 1;t) → P(x − a,t), and expand in a Taylo
